We showed where we paid all off. We showed we were paying bills on time. Write out your situation in a form of a letter. Let the lender know that you have saved money for a rainy day. You'll get that home - I just know you will.
Some banks are very very picky - so check out a couple of different ones.
Maybe a Credit Union too. I would disagree with adding additional debt only to have it appear as a good payment history. Accounts such as rent, utilities, and car insurance have generally been sufficient in situations similar to the one you described. I would make sure that each of these accounts is paid on time and contact a few lenders prior to making an offer. You can get a mortgage but you will pay a higher rate of interest due to the fact of your 'bad credit' history.