<PAGE>   1
                                                                    Exhibit 10.2

                             AGREEMENT OF AMENDMENT

         This AGREEMENT OF AMENDMENT (this "Agreement") dated as of November 27,
1998, among DAIRY MART CONVENIENCE STORES, INC. (the "Company"), BANKBOSTON,
N.A., successor by merger to BANK OF BOSTON CONNECTICUT, individually and as
Agent under the Credit Agreement (as hereinafter defined), and STATE STREET BANK
AND TRUST COMPANY ("State Street").

         WHEREAS, the Company, the banks and other financial institutions listed
on Schedule 1 thereto (collectively, with any banks or financial institutions
from time to time parties thereto, the "Banks"), and Bank of Boston Connecticut
("BOBC"), as agent of the Banks thereunder (in such capacity, the "Agent")
entered into a certain Credit Agreement dated as of April 24, 1996 (as amended,
the "Credit Agreement"); and

         WHEREAS, the Company, the Banks and the Agent wish to amend the Credit
Agreement as more fully set forth below.

         NOW THEREFORE, the parties hereto agree as follows:

                           I. AGREEMENT OF THE PARTIES

         1. Unless the context shall otherwise require, all capitalized terms
used herein without definition shall have the meanings assigned to them in the
Credit Agreement.

         2. Schedule I to the Credit Agreement shall be, and hereby is, amended
by deleting such Schedule I in its entirety and substituting the attached
Schedule I therefor.

         3. Section 1.1 of the Credit Agreement shall be, and hereby is, amended
by deleting the definitions of "Capital Expenditures", "Letter of Credit Rate",
"Minimum Consolidated Net Worth" and "Termination Date" in their entirety and
substituting in lieu thereof the following:

         "CAPITAL EXPENDITURES" any payment made directly or indirectly for the
         purpose of acquiring, constructing or improving fixed assets, real
         property or equipment which in accordance with GAAP would be added as a
         net debit (after giving effect to any credits) to the fixed asset
         account of the Person making such expenditure; PROVIDED, HOWEVER, that
         expenditures, which otherwise would constitute Capital Expenditures
         hereunder, funded by external sources such as bank financing, capital
         or operating leases, allowances from gas companies, etc. shall not be
         deemed to be Capital Expenditures for purposes of this definition.

         "LETTER OF CREDIT RATE" for each Letter of Credit, at any time, a rate
         per annum equal to the rate set forth below opposite the applicable
         ratio of Consolidated 

<PAGE>   2
         Indebtedness to Consolidated EBITDA for the period of four consecutive
         fiscal quarters ending on the FQED immediately preceding such time:
<TABLE>
<CAPTION>

          -------------------------------------------------------------------
                 Ratio of Consolidated
                   Indebtedness to 
          Level Consolidated EBITDA                     Letter of Credit Rate
          -------------------------------------------------------------------
          <S>                                                <C>  
            I Less than 3.00 to 1.00                              1.00%
          -------------------------------------------------------------------
           II Less than 3.50 to 1.00, but 
              greater than or equal to 
              3.00 to 1.00                                        1.25%
          -------------------------------------------------------------------
           IIILess than 4.00 to 1.00, but
              greater than or equal to 
              3.50 to 1.00                                        1.50%
          -------------------------------------------------------------------
           IV Less than 4.50 to 1.00, but 
              greater than or equal to
              4.00 to 1.00                                        1.75%
          -------------------------------------------------------------------
            V Greater than or equal to 
              4.50 to 1.00                                        2.00%
          -------------------------------------------------------------------
</TABLE>

         "MINIMUM CONSOLIDATED NET WORTH" (i) as at the FQED to occur on or
         about February 1, 1997: $7,500,000; (ii) as at the FQED to occur on or
         about April 30, 1997: $6,750,000; (iii) as at the FQED to occur on or
         about July 31, 1997: $7,900,000; (iv) as at the FQED to occur on or
         about October 31, 1997: $8,700,000; (v) as at the FQED to occur on or
         about January 31, 1998: $6,300,000; (vi) as at the FQED to occur on or
         about April 30, 1998: $4,800,000; (vii) as at the FQED to occur on or
         about July 31, 1998: $5,400,000; (viii) as at the FQED to occur on or
         about October 30, 1998: $5,850,000; and (ix) as at the FQED to occur on
         or about January 31, 1999: $5,600,000 and thereafter, $5,600,000 plus
         50% of Cumulative Consolidated Net Income earned after the FQED ending
         approximately January 31, 1999.

         "TERMINATION DATE" April 30, 2003, or such later date to which the
         Termination Date may be extended in accordance with subsection 2.15.


                                       2
<PAGE>   3


         4. Section 1.1 of the Credit Agreement shall be, and hereby is, amended
by adding the following definition of "Store Properties Held For
Sale/Leaseback":

         "STORE PROPERTIES HELD FOR SALE/LEASEBACK" as of any date, the book
value, as indicated on the Company's financial statements and determined in
accordance with GAAP, of any real or personal property owned by the Company and
held in anticipation of entering into a Sale/Leaseback Transaction with respect
thereto.

         5. Section 2.16 of the Credit Agreement shall be, and hereby is,
amended by deleting such Section 2.16 in its entirety and substituting in lieu
thereof the following:

         2.16 ANNUAL CLEAN-DOWN OF REVOLVING CREDIT LOANS. If at any time the
Company shall default in the observance or performance of any agreement
contained in Section 7, irrespective of whether such default is cured by the
Company or any action is taken by the Agent or any Bank, then, for a period of
at least 30 Consecutive Days during the Clean-Down Periods occurring during the
fiscal years set forth below, the aggregate principal amount of Revolving Credit
Loans outstanding at all times during such 30-day period shall not exceed the
amount set forth opposite the respective fiscal years:

<TABLE>
<CAPTION>
                                           Maximum Revolving Credit
            Fiscal Years                      Loans Outstanding
            ------------                   ------------------------

                 <S>                         <C>        
                     2001                        $18,000,000
                     2002                         12,000,000
                     2003                          6,000,000
</TABLE>

         6. Section 7.1(a) of the Credit Agreement shall be, and hereby is,
amended by deleting such Section 7.1(a) in its entirety and substituting in lieu
thereof the following:

         (a) ADJUSTED CONSOLIDATED INDEBTEDNESS TO CONSOLIDATED EBITDA. For any
         period of four consecutive fiscal quarters ending on any FQED set forth
         below, permit the ratio of (i) Adjusted Consolidated Indebtedness at
         the end of such period less Store Properties Held For Sale/Leaseback at
         the end of such period to (ii) Consolidated EBITDA for such period to
         be more than the ratio set forth opposite such FQED:

<TABLE>
<CAPTION>

--------------------------------------------------------------------------------
                    FQED                                               RATIO
--------------------------------------------------------------------------------
<S>                                                              <C> 
The FQED ending on or about February 1, 1997                        4.50 to 1.00
--------------------------------------------------------------------------------
</TABLE>

                                       3

<PAGE>   4

<TABLE>
<CAPTION>

--------------------------------------------------------------------------------
<S>                                                              <C> 
The FQED ending on or about April 30, 1997                          5.25 to 1.00
--------------------------------------------------------------------------------
The FQED ending on or about July 31, 1997                           3.80 to 1.00
--------------------------------------------------------------------------------
The FQED ending on or about October 31, 1997                        4.60 to 1.00
--------------------------------------------------------------------------------
The FQED ending on or about January 31, 1998                        5.00 to 1.00
--------------------------------------------------------------------------------
The FQED ending on or about April 30, 1998                          5.85 to 1.00
--------------------------------------------------------------------------------
The FQED ending on or about July 31, 1998                           5.65 to 1.00
--------------------------------------------------------------------------------
The FQED ending on or about October 30, 1998                        5.25 to 1.00
--------------------------------------------------------------------------------
The FQED ending on or about January 31, 1999                        4.65 to 1.00
--------------------------------------------------------------------------------
The FQED ending on or about April 30, 1999                          4.50 to 1.00
--------------------------------------------------------------------------------
The FQED ending on or about July 31, 1999                           4.35 to 1.00
--------------------------------------------------------------------------------
The FQED ending on or about October 30, 1999                        4.25 to 1.00
--------------------------------------------------------------------------------
The FQED ending on or about January 31, 2000                        4.25 to 1.00
--------------------------------------------------------------------------------
The FQED ending on or about April 30, 2000                          4.15 to 1.00
--------------------------------------------------------------------------------
The FQED ending on or about July 31, 2000                           4.10 to 1.00
--------------------------------------------------------------------------------
The FQED ending on or about October 30, 2000 and thereafter         3.90 to 1.00
--------------------------------------------------------------------------------
</TABLE>

         7. Section 7.1(b) of the Credit Agreement shall be, and hereby is,
amended by deleting Section 7.1(b) in its entirety and substituting in lieu
thereof the following:

         (b) EBITDA TO INTEREST EXPENSE. For any period of four consecutive
         fiscal quarters ending on any FQED set forth below, permit the ratio of
         (i) Consolidated EBITDA for the applicable period to (ii) Consolidated
         Interest Expense for such period to be less than the ratio set forth
         opposite such FQED:

<TABLE>
<CAPTION>
--------------------------------------------------------------------------------
                  FQED                                                RATIO
--------------------------------------------------------------------------------
<S>                                                            <C> 
The FQED ending on or about February 1, 1997                       2.00 to 1.00
--------------------------------------------------------------------------------
The FQED ending on or about April 30, 1997,                        1.60 to 1.00
--------------------------------------------------------------------------------
</TABLE>

                                       4

<PAGE>   5

<TABLE>
<CAPTION>

<S>                                                            <C> 
July 30, 1997 and October 31, 1997                                 
--------------------------------------------------------------------------------
The FQED ending on or about January 31, 1998                       1.60 to 1.00
--------------------------------------------------------------------------------
The FQED ending on or about April 30, 1998                         1.40 to 1.00
--------------------------------------------------------------------------------
The FQED ending on or about July 31, 1998                          1.45 to 1.00
--------------------------------------------------------------------------------
The FQED ending on or about October 30, 1998                       1.60 to 1.00
--------------------------------------------------------------------------------
The FQED ending on or about January 31, 1999                       2.00 to 1.00
--------------------------------------------------------------------------------
The FQED ending on or about April 30, 1999                         2.00 to 1.00
--------------------------------------------------------------------------------
The FQED ending on or about July 31, 1999                          2.00 to 1.00
--------------------------------------------------------------------------------
The FQED ending on or about October 30, 1999                       2.10 to 1.00
--------------------------------------------------------------------------------
The FQED ending on or about January 31, 2000                       2.10 to 1.00
--------------------------------------------------------------------------------
The FQED ending on or about April 30, 2000 and                     2.25 to 1.00
thereafter
--------------------------------------------------------------------------------

</TABLE>


         8. Section 7.1(c) of the Credit Agreement shall be, and hereby is,
amended by deleting Section 7.1(c) in its entirety and substituting in lieu
thereof the following:

            (c) FIXED CHARGE COVERAGE. For any period of four consecutive fiscal
            quarters ending on any FQED set forth below, permit the ratio of (i)
            Consolidated EBITDAR minus the amount of any federal, state and
            local income taxes levied by a Governmental Authority on the
            revenues of the Company which are actually paid by the Company or
            its consolidated Subsidiaries in cash during such period, to (ii)
            Consolidated Interest Expense, plus all principal payments required
            to be made during the period on account of any Consolidated
            Indebtedness, plus the amount of any Consolidated Rent Expense
            during the period, to be less than the ratio set forth opposite such
            FQED:

<TABLE>
<CAPTION>

--------------------------------------------------------------------------------
                  FQED                                                RATIO
--------------------------------------------------------------------------------
<S>                                                             <C> 
The FQED ending on or about January 31, 1998                       1.25 to 1.00
--------------------------------------------------------------------------------
The FQED ending on or about April 30, 1998                         1.05 to 1.00
--------------------------------------------------------------------------------

</TABLE>

                                       5
<PAGE>   6

<TABLE>
<CAPTION>

--------------------------------------------------------------------------------
<S>                                                              <C> 
The FQED ending on or about July 31, 1998                          1.15 to 1.00
--------------------------------------------------------------------------------
The FQED ending on or about October 30, 1998                       1.20 to 1.00
--------------------------------------------------------------------------------
The FQED ending on or about January 31, 1999 and                   1.25 to 1.00
thereafter
--------------------------------------------------------------------------------

</TABLE>


         9. Section 7.8 of the Credit Agreement shall be, and hereby is, amended
by deleting Section 7.8 in its entirety and substituting in lieu thereof the
following:

         7.8 LIMITATION ON CAPITAL EXPENDITURES. Make or commit to make (by way
         of the acquisition of securities of any Person or otherwise) any
         Capital Expenditures (excluding any expenses incurred in connection
         with normal replacement and maintenance programs properly charged to
         current operations) in the aggregate for the Company and its
         Subsidiaries during any of the fiscal years of the Company set forth
         below in an amount in excess of that set forth opposite such fiscal
         year:
<TABLE>
<CAPTION>

                                                    Maximum Capital
                  Fiscal Year                        Expenditures
                  -----------                        ------------

                   <S>                           <C>        
                      1997                           $28,500,000

                      1998                           $34,500,000

                      1999                           $35,000,000

                  Each fiscal year
                  thereafter                         $25,000,000
</TABLE>

         PROVIDED, however, that in the event the Company and its Subsidiaries
         shall make Capital Expenditures in any fiscal year of less than the
         maximum amount set forth above, an amount equal to the difference
         between such maximum amount and the actual Capital Expenditures made
         during such fiscal year shall be available (in addition to the maximum
         amount stated above) to the Company for Capital Expenditures during
         (but only during) the immediately following fiscal year.

         10. Except as specifically amended or modified by this Agreement, all
terms and conditions set forth in the Credit Agreement and the Security Document
remain in full force and effect.

         11. This Agreement shall be governed by and construed in accordance
with the laws of the State of Connecticut.

                                       6

<PAGE>   7

         12. The Company shall reimburse the Agent for the fees and expenses of
its counsel in connection with this Agreement.

         13. This Agreement may be executed in two or more counterparts, each of
which shall constitute an original but all of which when taken together shall
constitute but one agreement.

                  II. REPRESENTATIONS AND WARRANTIES OF COMPANY

         The Company represents and warrants as follows:

         1. The representations and warranties set forth in the Credit Agreement
are true and correct in all material respects on the date hereof as if made on
the date hereof.

         2. After giving effect to this Agreement, no Event of Default has
occurred and is continuing, and no event or condition has occurred or exists
which would constitute an Event of Default but for the giving of notice or
passage of time or both.

         3. There has been no material adverse change in the assets,
liabilities, operations or financial condition of the Company or any Designated
Subsidiary.


                        III. CONDITIONS TO EFFECTIVENESS

         This Agreement shall be effective upon satisfaction of the following
conditions:

         1. The Agent shall have received counterparts of this Agreement which,
when taken together, bear the signatures of all parties hereto.

         2. The representations and warranties of the Company set forth in
Section II of this Agreement shall be true and correct in all respects.

         3. The Agent shall have received payment in immediately available funds
of an amendment fee in the amount of $150,000.

                                       7

<PAGE>   8


         IN WITNESS WHEREOF, the parties have caused this Agreement to be
executed and delivered as of the date first above written.

                                     DAIRY MART CONVENIENCE STORES, INC.


                                     By:_______________________________________
                                        Name:
                                        Title:


                                     BANKBOSTON, N.A., successor by merger to
                                     BANK     OF     BOSTON     CONNECTICUT, 
                                     Individually and as Agent


                                     By:_______________________________________
                                        Name:
                                        Title:


                                     STATE STREET BANK AND TRUST COMPANY


                                     By:_______________________________________
                                        Name:
                                        Title:

                                       8

<PAGE>   9

                                   SCHEDULE I

                             COMMITMENTS; ADDRESSES


<TABLE>
<CAPTION>
                                                              Commitment
                                                               $ Amount
--------------------------------------------------------------------------------
<S>                                                       <C>        
BANKBOSTON, N.A.                                              $17,500,000
100 Pearl Street, 5th Floor
Hartford Corporate Banking
Hartford, Connecticut 06103

    ATTENTION:  Scott S. Barnett, Vice President
    TELECOPY NO.: 860/727-6565


State Street Bank and Trust Company                           $12,500,000
Goodwin Square Building
225 Asylum Street
Hartford, CT 06103

    ATTENTION:  Karen Booth, Vice President
    TELECOPY NO.:  860/244-1810

TOTALS:                                                       $30,000,000

</TABLE>