You will need to 'Forget' the AP from the software currently on your existing install then re-adopt it in the new setup.
No you don’t.
If you take a backup of the current controller and then restore that backup when setting up the new controller then the access points will connect to the new controller without a problem.
Surprising that they said that as its trivial to move the controller from one system to another. Certainly that’s easier then starting from scratch with a new controller and doing the setup again.