It is possible for you to get Chinese visa from Chian embassy in Bangkok, but it is not 100%. It is easiest to go to the China embassy in person to submit the application, no appointment is required, it is easy for USA citizens to get a China tourist visa in Bangkok, if you apply for it through a travel agency, you have to pay for extra fee for its service. You can't fly from Bangkok to Phuket with your US driver's license. It takes 4 working days for visa processing, therefore, your original passport should be left at the visa office of the embassy for 4 working days.