Your credit rating is affected for 5 years after you repay the debt so in the case of a large debt it could be years before you have a clear record.
Handing back the keys does not mean the lender will sell the property any quicker than you but you will be liable for the debt until the house is sold and the debt satisfied. Also you dont have any say over the selling costs, choosing an estate agent, negotiating a rate, agreeing an acceptable selling price. Also if on the market for a long period hte property will become rundown. Rather than go down that road would you consider the following:
-get a local letting agent to put in writing the chances of letting and the amount of rent that might be recd.
-get a local estate agent to put in writing what they think the property may sell for and how long it may take to sell it
-bring these documents along with an affordabe payment proposal to your lender which will show you are proactive about your debt and trying to resolve it
-ask for an interest only on your mortgage(s) for a year to see if market lifts, then they can review
-advertise invest. property to let
-if you have grown up kids let it to them if they can afford it
-try letting your own home and move in to investment property
-maximise your income by letting a room in your own home
-ask your local authority about the Residential Assesement Scheme (RAS) where the local authority pay the rent directly to you for a defined period. Rent may be a little lower but no voids
best of luck, hope it goes well for you
guby