Yeah i've heard of a couple of situations where people have managed to get a z visa processed in China (both in Beijing) but i wouldn't count on it.

Finding someone who knows the system to negotiate with the school is definitely a good idea. I managed to convince a school that i worked at to not cancel the RP of another teacher who was leaving as part of a negotiation that took place in regards his departure. I also managed to do the same with the last job i left mid contract, although i did leave that position on good terms with those in charge and therefore it made the process a lot easier.
Also if you can find a new job before you leave then you can get your new employer to do the negotiating on your behalf. Clearly they have a vested interest in you being able to start your new job sooner rather than later and at lesser hassle and cost than if you have to leave the country and start the visa process all over again. They will also be in the best position to know exactly what they need from your former employer in terms of documents, visas,etc.