Your NEW SCHOOL, if they are competent and licensed to hire you, can manage the process of acquiring a new residence permit while your existing residence permit is still in force and sitting in your passport. They will, however, require the cancellation of your Foreign Expert Certificate. You are able to cancel that yourself, if you can get your hands on it, and perhaps even if you have a copy.

The issue here is the 'release letter' from your old school which your new school may require and the PSB in most provinces now does require.

Hopefully your new school has the motivation and guanxi to communicate with your old school to be reasonable about your release. If the old school still wants to F... with you after they speak to your new school (or if you don't want them to speak to each other)... the old school can find any number of reasons to deny you the release letter.

I have never heard of anyone getting a new residency permit while on a temporary visa extension. But that doesn't mean it doesn't happen. Good luck.