If your host is doing this without significant advance notice (say, 2-3 months, minimum) then you need to change hosts. Period. Good hosts don't do this to their clients on short notice.
The changeover might go perfectly well.
But you should still change hosts.
That said, your site doesn't sound very complicated, and I'd expect it would go well. But, who knows? I don't know your host. I've had this go very, very badly for me in the past.
I'd hope there will be some way to test the new account before the old one is shut off.
