Got hit with something that I guess is common, but I didn't expect. The sellers want to stay in the home for 7 days past close.
That is a very nasty last minute high risk change. It happens often, but that doesn't reduce the risks involved.
For example: are you prepared to purchase a landlord's insurance policy to cover a catastrophic loss, such as if they burn the house down in the 7 day period while you own it and are responsible? I seriously doubt that your insurance you are purchasing as the property owner would/is going to cover such a loss by a tenant. Their insurance would not cover as they are no longer a property owner.
Like it or not, you are being pushed into being a landlord with all those added rights and responsibilities. I realize it is a stupid question but can they cook meth on the property during the 7 day period? If their diesel tank ruptures and spills everywhere, are they, or you, responsible for property cleanup? What if their dog bites someone? What if after 7 days they say another hitch has developed and now they need 90 days? And it goes on and on.
My temptation would be to say you are not going to close escrow until you can have possession and they have moved out. But I don't know all of the details of your transaction and laws of your state.
If it were me and I did allow them to hold over, I'd want some language where they are fully responsible for damages, etc. until they vacate-- and until you do a walkthrough after they have vacated and you approve the condition.
A mid-strategy is to let them stay, have a signed agreement that they do not get the sale proceeds until they: a) vacate, and b) you approve the condition after they are gone. And something like you can use their part of the sale proceeds to correct any damage that might occur.
I'm not intending to give legal advice as I am not qualified to do that. So, you might want to consult a real estate attorney about them staying past the close of escrow. I realize this is unwanted, but it would be terrible if you were on the hook to pay the mortgage on a property someone else destroyed, and which you could never even move into.
Good luck!