Visa extension or renewal

Hi. I'm an American teacher in Shanghai and my residence permit expires on June 22nd but my contract expires on June 30th. My current school is not willing to cooperate with me on extending my permit until June 19th or June 20th.

I was offered another job and school B would like paperwork from school A. Now, what can I legally do on obtaining a new visa if school A is not willing to cooperate until June 19th or June 20th?