Three ways out of ownership are to: short sale the house, ask the bank to accept a Deed In Lieu of Foreclosure, or allow the house to be foreclosed.
Generally speaking, mortgage payments which are less than 30 days late are not reported "late" on your credit report although you will pay a late fee. A payment is 30 days late when two payments become due (e.g. payment due in January - not made until February which then means payment for February is also due.)
Depending upon how late your tenants are paying rent, you might want to consider some very strongly worded letters to them or even consider eviction according to the laws of your state. This "may" cause them to accept their responsibility in a timely manner and prevent you distress.
Another alternative to the "late" payment cycle is to get one month ahead. Despite this sounding ridiculous given your situation, you might want to take an interest free cash advance from one of your credit cards. Doing so may breathe "life" into your financial dilemna and prevent further destruction of your credit.
Of course you should first consult an attorney in all matters BEFORE making a decision.