As aChineseBostonian pointed out, you can

just go ahead and drive, with your American driver's license. The Chinese police have more than enough to keep themselves busy than randomly check your driver's license. Even if you get into an accident (well, you will need to pay for the accident if it is your fault, but that is what you should expect), you can simply show the police your American license and say it is an international driver's license, or simply say you left it at home.

I have a friend. He has been caught numerous times driving (he has a license but just left it home) due to things like going through red lights. But he has always been able to talk his way out, with stories like "my wife is delivering a baby in a hospital", and give the police a business card (that is right, he did not carry an ID either).