It's really hard for Americans to get things done in China, no matter how well-meaning they are. If you are an established scholar, you may be able to get a Fulbright teaching fellowship. The Hopkins-Nanjing Center also places teachers of Economics, Government, and Law at their center. If you can do this, you will experience China in style. The Fulbright stipend is about 10 times what Foreign Experts are paid, you only have to teach about 6 hours a week, the U.S. government pays for your students' books, they pay for you to take Chinese language classes before you go, and you get to hang out at the embassy, etc.
Many universities have exchanges where they send faculty to teach for a few months.
Then there's research funding and international organizations like World Bank, Rockefeller Foundation, UNDP, Ford Foundation, etc. I would like to know how to get these funds myself, so I won't give you any advice. There are no hard and fast qualifications for getting a job teaching English in China. These days there is a huge demand, and there are schools that will take anybody who speaks English. If you have a degree in English or from an English-speaking country, you should be able to find something.