When you buy a home to put family in or occupy tenants really can't fight that. I would suggest you speak to your attny.
You need 30 days notice for some people ect. it is complicated. Let the current owners deal with it. Your agent or the sellers agent could give them some rental listings and some help, but if they are not good tenants I personally wouldn't get involved.
You're right. It's not your job to find them an apt. If the tenant's lease has expired then your agent and attorney should get on the ball and earn their pay!! As I mentioned earlier - your homeowner's insurance may not begin providing coverage if they are there and YOU are not! AND you don't want to risk them damaging the place. Boy I wish you were in NC. We wouldn't tolerate this kind of nonsense.
postpone closing! the seller or their agent should have dealt with this issue. I am sure there is a protection in the contract, to postpone without risk of lose. The last thing you want is to have to do the eviction. Probably very expensive, and timely...do you have an agent working for you? Light a fire under their tail get them working!
The current owner needs to serve them with a letter to terminate the lease. You need to follow your states guidelines. If they fail to vacate they are considered a tenant in holdover and must be evicted.