So far you have been professional.

Your current school is protecting their own interests. I recommend you have a chat with the Senior FAO manager:

"If you are able to provide the release document and cancel the FEC on line by next Wednesday I will be able to complete the process for my new employment. I understand that this is an inconvenience for your staff.

I want to stay through the end of this semester to teach all my classes at your school and to provide you with all your students grades.

If, however, I am unable to transfer my residency permits in China, I need to return home ASAP to look for a new job teaching in my hometown. The sooner I return home the better chance I have to find a job teaching there in September.

I hope that you are able to help me; I DO want to stay to finish my teaching at your school this semester."

They will get the message.