What is the process to change my visa and work permit when moving to a new school in Thailand?

extension of stay COULD be canceled several days earlier at some immigration office. If you can, applied cancellation in advance and you can apply for a new extension on the right days in the right place. You had better check your current immigration office if they let you do.

Both of your employers also need to work accordingly to cancel your current work permit and apply a new one along the right timeline. but this part is out of the scope of this group.

You can stay until 26th Sep on amnesty(aka overstay without being fined or charged). But you cannot get workpermit (that you need for working legally) when you are on amnesty.

Your new employer should be able to advise on this, especially since 1. you resigned from your previous job, and 2. you are relocating... TO TAKE UP EMPLOYMENT WITH THEM.

Every foreigner needs a Work Permit and a visa or Extension of Stay, that allows a foreigner to work inside Thailand in combination with that work permit,

You do not need a visa to be inside Thailand is the worst advice you can ever get,