I am pretty certain that you can have it renewed without having to leave the country. You will need the release letter and you may need the reference letter from your current/previous employer. The fact that the schools that you are going to do not know should raise a warning flag. I believe they are either new to hiring foreigners (could be a headache for you) or they are used to duping new people to China and having them run away... so they aren't used to duping people who are already in China (since they are already the wiser).

I would check into it. Either way, you should be fine as long as you have both of those documents.

TEO