Most refinances are done to "get out" of a mortage and get a better deal. If this is not an option due to lack of equity, or credit; it is now more possible than ever before to actually renegotiate with your current lender. Lenders are currently acquiring many more properties than they want to hold due to foreclosures. This provides a huge incentive to make it possible for their current customers to be able to afford their mortgage payment and not have to sell or face foreclosure.
Yes, pay the mortgage in full.
Another option is to deliver the deed (the rights) to the house "in lieu of" (instead of) foreclosure. The lender would have to agree to this provision. In this instance, the consumer is handling over the property and all his equity in exchange for the debt being cancelled. Obviously, a mortgage lender would only consider such an option if it was in their best interests.
Deed in lieu of foreclosure is considered a derogatory credit notation, although not as bad as foreclosure itself, and will remain on a consumers' credit report for 7 years.
Just be aware that if a deed-in-lieu is approved, that the lender does report it to the IRS as a 'forgiven debt' so there are tax consequences. I believe that it�s considered income to you, and therefore you have to pay taxes on it. The advantage to the mortgagor of a DIL vs a foreclosure is that it does look somewhat better on your credit report, and you don't have to worry about the lender seeking a deficiency judgment against you in the event that the property resells for less than the mortgage amount.
You should be aware that I'm not an attorney, but I did work in mortgage servicing for a numbers of years.