I have two problems with this.
The 3rd problem is that you dealing with an organisation that has no vested interest in being helpful now that you are on the way out. They seem to be playing the usual 'run it around the block until it goes away' game.
I was faced with that 3 jobs ago and it was only solved by keeping in the face of the FAO responsible.
The key points are:
1. You must get the document before you leave because they will all bugger off for the holidays no matter what they say about the office being open
2. It is best for you to give them a template for the Leaving Clearance document. The best example I have seen is on the Hunan State Authority for Foreign Experts site, and you shouls be able to download it from there. It is not a basic document.
3. Identify the FAO person who is going to issue the document to you in person (as put it in your hand) and persecute them. Hold them personally responsible for solving the problem - if they try to duck-shove then tell them you could not give a fuck about the details of their system ... that is the FAO's problem to solve. Get a deadline out of them that suits you ... and leaves at least 3 days to sort out if they do not deliver on time.
4. the document needs to be on red letter head and have a red stamp. Do not let them try to give you anything less.
The Nanjing thing sure complicates things but nailing the FAO (at the school you are teaching at) to the wall seems to be the only way to go. All they have to do is email your template (with the details filled in) to Nanjing and get them to Express Mail back the stamped document on letterhead.
Stay strong!