for Thailand driver license you need address in Thailand. If you are US passport, just got embassy and you do an affidavit with your thailand address on it. It cost 50usd. You will need to book an appointment about 3 weeks out.
you got a resident cert from immigration? You should not be able to on visa exempt or TR60 because you have to file 90 day report first before a resident cert can be issued...which you would not do as those visa options do not allow you stay longer than 90 days.
Yes you can a license in Thailand, even on a tourist visa. Since you don't have one already, it will take 2 days.
I recommend you use a service like fat boys to help stream line the process for you.
The biggest issue you might have is getting a resident certificate on short notice. As a tourist you can't get from immigration, you can use you embassy. Different embassy have different requirements. The US embassy you just do an affidavit saying what your address is. The problem is appointment are typically booked out several weeks.