Inquiry into residency cards and application for visa to China?

I am an American citizen living and working in Jordan - this is my fourth year. I am employed at a private international school. The beginning of fall term our residency cards are renewed. The process begins in August and sometimes takes up to three months to complete. I am scheduled to visit Shanghai China for school. I have been to the Chinese Embassy numerous times, but because my current cards have expired they will not issue the visa? They told me to visit the local police and immigration dept. To get official papers? I visited both bit they could not help me. I am scheduled to visit the US Embassy. IAA has provided proof of my employment, contract, etc. even a letter from the school indicating IAA is actively working on the renewal of my residency cards. I have the required invitation letters from China as well, but still the Chinese Embassy in Jordan will not issue me the visa? What are my options now as an American citizen? Any insight, direction or guidance would be much appreciated.