Has anyone sold and bought a house and had the settlement on the same day?
You don’t need to surrender the keys prior to closing, regardless of what the other estate agent says. However at closing the place you are selling must be ready for the new occupant unless you have a contractual stipulation giving you additional time to shift your gear. If your buyer is not willing to grant a concession, be prepared for two moves. There’s no way around that. I’ve bought and sold several homes with possession 7 or 14 days after closing, with an adjustment to the buyer for rent for the home while they await possession. This is common, though your buyer is under no obligation to accept this offer.
