I agree with sdrealtor – bird in the hand with the current buyer, I would work with them on extending the escrow front BUT politely explain that you cannot enter into a short term rental that it is too fraught with potential conflict/issues.

Do you have the right to any involvement, verification from their lender/broker? I would extend escrow but with some “check points” to make sure things keep moving along.

ALSO – what is your living situation? If you have not moved out of the house and are going to have double rent anyhow – I would consider staying and/or leaving sufficient furnishings in case you DO need to go back on the market. Furnished homes show so much better.