In Texas, it's possible to do a “deed in lieu of foreclosure" if your mortgage company will agree. But deeds in lieu of foreclosure are not done as much in Texas as in other U.S. states.
This is because Texas has “non-judicial” foreclosure procedures, at least for purchase-money mortgages. It is therefore relatively quick and easy for mortgage lenders to foreclose on houses in Texas.
Also, they can get a better title to the house if they foreclose, since most intervening liens or clouds on your title can be cut off by a mortgage foreclosure.
But it may benefit you somewhat to do a “short sale” if there is no equity in the property and no way to do a conventional sale. Typically in a short sale, the mortgage company (or companies if you have more than one mortgage) agree to accept less than the full amount owed in full satisfaction of what you owe them.
Not all short sales are successful; sometimes the mortgage company, or some other party that holds a lien on the property like a second lien or a homeowner association, will not agree to “take a haircut,” i.e. agree to accept less than the full amount that they are owed. So the short sale may not work.
I have seen short sales completed where you (the seller) signed a promissory note, to pay money to the mortgage company after the sale for the rest of the money that was owed. I don’t know why you would agree to do that, if you are having financial problems and don’t have the money to pay it, barring some unusual circumstances.
After all, you have the option of walking away if you want to. But if you walk away, there is a chance that the mortgage company could try to sue you for any "deficiency balance" due after a foreclosure. Watch the video below from the government's "Making Home Affordable" program.
Your first step to complete a short sale would likely be to find a real estate agent or company that handles short sales, to list the property for sale and to start negotiations with the mortgage company.
Your real estate agent or real estate company could also be paid out of the proceeds of sale. If you don't have someone already to do that for you, we can refer you to companies that do that.