Be careful what you sign. New agreements signed are after the bankruptcy and could create new obligations. You can do it but be careful. The best advice is to have an attorney review the documents before you sign them
No. Accepting the cash for keys offer shouldn't affect your bankruptcy discharge. Even after a bankruptcy is filed, the bank still has to file a state court foreclosure action against you to take the home. There is no reason for you to short sell the property, since you have filed for bankruptcy and all debt owed to the bank can no longer be collected against you. You may continue to occupy the home until legally evicted through the foreclosure process, but you may prefer to move sooner with the cash for keys offer, which will benefit both you and the bank, since it will cost them less to recover the house.
No, the cash for keys should have no affect on your bankruptcy discharge. You should consider whether the amount they are willing to pay you offsets whatever rent you would have to pay in the time between now and whenever they would eventually get around to foreclosing if you do not do the short sale and the cash for keys. Effectively you could continue living there for your HOA dues, homeowners insurance and taxes which could be less than you'd have to pay for rent.
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I feel it is important to point out that there is a big difference between signing the bankruptcy paperwork saying that someone "intends to surrender" property and the actual surrender of the property. Some bankruptcy trustees are arranging for the "short sale' of underwater properties when the debtors say they intend to surrender & taking the "cash for keys" money to pay the debts. Be sure your case is actually closed and that your trustee doesn't have his/her own plans for the property. Hope this perspective helps!