I know a FT, who ran away from his school in Gansu. He told the FAO about his plan to leave and got a release letter from the Foreign Affairs Office of that particular college. Shortly after, he got new job in another province and extended his work-visa there.
You need that RELEASE LETTER from your present school, if you want to teach somewhere else in China without any problems. Otherwise, another FAO from a different school cannot help you out.
Do you have a good relationship with your FAO ? What will you tell him ? Why do you want to leave his school ? ( some problems in your family at home, a bad health because of the climate in that specific province,... )