<PAGE>   1
                                                                   EXHIBIT 10.36

                               SIXTH AMENDMENT TO
                      AMENDED AND RESTATED CREDIT AGREEMENT


                  This Sixth Amendment to Amended and Restated Credit Agreement
(this "Amendment") dated as of November 19, 1998 is among COSTILLA ENERGY, INC.,
a Delaware corporation (the "Borrower"), the banks named on the signature pages
hereto (together with their respective successors and assigns in such capacity,
the "Banks"), BANKERS TRUST COMPANY, as agent for the Banks (together with its
successors and assigns in such capacity, the "Agent") and UNION BANK OF
CALIFORNIA, N.A., as co-agent for the Banks (together with its successors and
assigns in such capacity, the "Co-Agent").

                              PRELIMINARY STATEMENT

                  A. The Borrower and the Bank Group have entered into that
certain Amended and Restated Credit Agreement dated as of August 28, 1997 as
amended by that certain First Amendment to Amended and Restated Credit Agreement
dated as of December 30, 1997, that certain Second Amendment to Amended and
Restated Credit Agreement dated as of January 14, 1998, that certain Third
Amendment to Amended and Restated Credit Agreement dated as of February 26,
1998, that certain Fourth Amendment to Amended and Restated Credit Agreement
dated as of March 24, 1998 and that certain Fifth Amendment to Amended and
Restated Credit Agreement dated as of June 30, 1998 (as so amended, the "Credit
Agreement").

                  B. The Borrower and the Bank Group desire to further amend the
Credit Agreement as set forth herein.

                  NOW THEREFORE, in consideration of the foregoing and the
mutual agreements set forth herein, the parties agree as follows:

                  Section 1. Definitions. Unless otherwise defined in this
Amendment, each capitalized term used in this Amendment has the meaning assigned
to such term in the Credit Agreement.

                  Section 2. Amendments. The Credit Agreement is hereby amended
as follows:

                  a. Section 2.04(d) of the Credit Agreement is hereby amended
         in its entirety to read as follows:

                           "(d) So long as any of the Commitments are in effect
                  and until payment in full of all Loans hereunder, effective on
                  or about March 15 and September 15 of each year commencing
                  March 15, 1999 (each being a "Scheduled Redetermination
                  Date"), the Agent with the approval of all of the Banks or the
                  Majority Banks, as applicable, shall redetermine the amount of
                  the Borrowing Base in accordance with Section 2.04(b). If, on

<PAGE>   2



                  or before December 31, 1998, the Borrower has not consummated
                  the acquisition of the Oil and Gas Properties and related
                  assets from Pioneer Natural Resources USA, Inc. ("Pioneer") as
                  contemplated under that Purchase and Sale Agreement between
                  Pioneer and the Borrower, dated September 4, 1998, the Agent
                  with the approval of all of the Banks or the Majority Banks,
                  as applicable, shall redetermine the amount of the Borrowing
                  Base in accordance with Section 2.04(b) (the "Pioneer
                  Redetermination"). In addition, at any time after the first
                  scheduled Redetermination Date, (i) the Borrower may request a
                  redetermination of the Borrowing Base on its own initiation at
                  any time in connection with a proposed acquisition of Oil and
                  Gas Properties with a fair market value in excess of
                  $10,000,000 and at one additional time during any consecutive
                  twelve (12) month period (each redetermination of the
                  Borrowing Base under this sentence, whether initiated by the
                  Borrower or the Agent, being referred to herein as an
                  "Unscheduled Redetermination"), and (ii) the Agent may
                  initiate an Unscheduled Redetermination of the Borrowing Base
                  at any time as the Agent may so elect; provided, however, that
                  the Agent may initiate only one such Unscheduled
                  Redetermination during any consecutive twelve (12) month
                  period (each being an "Unscheduled Redetermination Date"). Any
                  Unscheduled Redetermination of the Borrowing Base on an
                  Unscheduled Redetermination Date shall be in accordance with
                  Section 2.04(b)."

                  b. Section 5.10(b) of the Credit Agreement is hereby amended
         in its entirety to read as follows:

                           "(b) With respect to the Pioneer Redetermination
                  contemplated by the second sentence of Section 2.04(d) or any
                  Unscheduled Redetermination of the Borrowing Base initiated
                  under Section 2.04(d), the Borrower shall furnish to the Bank
                  Group a Reserve Report prepared by or under the supervision of
                  the chief engineer of the Borrower (or upon the request of the
                  Agent prepared by independent petroleum engineers acceptable
                  to the Agent) covering the Borrower's Oil and Gas Properties.
                  Such Reserve Report shall be prepared in accordance with the
                  procedures set forth in Section 5.10(a), shall contain such
                  other information as the Agent may reasonably request and
                  shall have an "as of date" as requested by the Agent. For any
                  Unscheduled Redetermination of the Borrowing Base initiated by
                  the Agent under Section 2.04(d), the Borrower shall provide
                  such Reserve Report as soon as possible, but in any event no
                  later than 30 days following the Borrower's receipt of notice
                  of such Unscheduled Redetermination from the Agent. With
                  respect to the Pioneer Redetermination, the Borrower shall
                  provide such Reserve Report on or before January 31, 1999."


                                       -2-

<PAGE>   3



                  c. Section 6.07 of the Credit Agreement is hereby amended in
         its entirety to read as follows:

                           "Section 6.07. Sales of Properties. The Borrower will
                  not, and will not permit any of its Subsidiaries to, sell,
                  transfer, assign, farm-out, lease or otherwise transfer or
                  dispose of any Properties other than (a) sales of Hydrocarbon
                  production in the ordinary course of business and sales of
                  obsolete or worn-out equipment in the ordinary course of
                  business, (b) sales or transfers of Properties by any of the
                  Borrower's wholly-owned Subsidiaries to the Borrower or any
                  such other wholly-owned Subsidiary, (c) the sale of the Concho
                  1998 Properties for a purchase price of $8,466,000 (as such
                  purchase price may be adjusted under the terms of the Concho
                  1998 Purchase Agreement), and (d) any other sale of Properties
                  sold at fair market value, so long as the aggregate Net
                  Proceeds for all such sales made under this subclause (d)
                  during the period between each redetermination of the
                  Borrowing Base does not exceed $1,000,000."

                  d. The following defined terms are hereby added to Annex A of
         the Credit Agreement in their appropriate alphabetical order:

                           ""Concho 1998 Properties" means the Borrower's Oil
                  and Gas Properties and other related assets described as the
                  "Assets" in the Concho 1998 Purchase Agreement.

                           "Concho 1998 Purchase Agreement" means that certain
                  Purchase and Sale Agreement dated as of November 2, 1998, by
                  and between the Borrower and Concho Resources, Inc., a
                  Delaware corporation, as amended by that certain letter
                  agreement dated November 18, 1998."

                  Section 3. Ratification. The Borrower hereby ratifies and
confirms all of the Obligations under the Credit Agreement (as amended hereby)
and the other Loan Documents. All references in the Loan Documents to the
"Credit Agreement" shall mean the Credit Agreement as amended hereby and as the
same may be amended, supplemented, restated or otherwise modified and in effect
from time to time in the future.

                  Section 4. Effectiveness. This Amendment shall become
effective in accordance with the terms of the Credit Agreement subject to the
condition precedent that the Borrower shall have delivered to the Agent a
complete copy of the fully executed Concho 1998 Purchase Agreement. This
Amendment may be executed in separate counterparts, all of which constitute one
and the same instrument.

                  Section 5. Representations and Warranties. The Borrower hereby
represents and warrants to the Bank Group that (a) the execution, delivery and
performance of this Amendment has been duly authorized by all requisite
corporate action on the part of the Borrower, (b) each of the Credit Agreement
(as amended hereby) and the other Loan Documents to which it is a party
constitutes a valid and legally

                                       -3-

<PAGE>   4



binding agreement enforceable against the Borrower in accordance with its terms
except, as such enforceability may be limited by bankruptcy, insolvency,
reorganization, moratorium, fraudulent transfer or other similar laws relating
to or affecting the enforcement of creditors' rights generally and by general
principles of equity, (c) the representations and warranties by the Borrower
contained in the Credit Agreement as amended hereby and in the other Loan
Documents are true and correct on and as of the date hereof in all material
respects as though made as of the date hereof, (d) no Default or Event of
Default exists under the Credit Agreement (as amended hereby) or any of the
other Loan Documents.

                  Section 6. Choice of Law. This Amendment shall be governed by,
and construed in accordance with, the laws of the State of New York.

                  Section 7. Final Agreement. THE CREDIT AGREEMENT (AS AMENDED
HEREBY) AND THE OTHER LOAN DOCUMENTS REPRESENT THE FINAL AGREEMENT BETWEEN THE
PARTIES AND MAY NOT BE CONTRADICTED BY EVIDENCE OF PRIOR, CONTEMPORANEOUS OR
SUBSEQUENT ORAL AGREEMENTS OF THE PARTIES.
THERE ARE NO ORAL AGREEMENTS BETWEEN THE PARTIES.

                  IN WITNESS WHEREOF, the parties hereto have caused this
Amendment to be executed by its officers thereunto duly authorized as of the
date first above written.


                                           COSTILLA ENERGY, INC.


                                           By:  /s/ 
                                               --------------------------- 
                                           Name:
                                                --------------------------
                                           Title:
                                                 -------------------------



                                       -4-

<PAGE>   5



                                           BANKERS TRUST COMPANY,        
                                             as Agent and Bank


                                           By:   /s/ 
                                              ------------------------------ 
                                           Name:                            
                                                ---------------------------- 
                                           Title:                           
                                                 --------------------------- 

                                           UNION BANK OF CALIFORNIA, N.A.,
                                              as Co-Agent and Bank


                                           By:   /s/                
                                              ------------------------------
                                           Name:                            
                                                ---------------------------- 
                                           Title:                           
                                                 --------------------------- 


                                           By:   /s/                 
                                              ------------------------------
                                           Name:                            
                                                ---------------------------- 
                                           Title:                            
                                                 ---------------------------

                                           DEN NORSKE BANK ASA,
                                              as Bank


                                           By:   /s/                
                                              ------------------------------
                                           Name:                            
                                                ---------------------------- 
                                           Title:                            
                                                 ---------------------------

                                           By:   /s/                 
                                              ------------------------------
                                           Name:                            
                                                ----------------------------
                                           Title:                           
                                                 ---------------------------

                                           WELLS FARGO BANK (TEXAS), N.A.,
                                              as Bank


                                           By:   /s/                
                                              ------------------------------
                                           Name:                            
                                                ---------------------------- 
                                           Title:                            
                                                 ---------------------------


                                       -5-