No, but I would highly recommend you get a license, especially if you’re you are going to be doing most of the driving, and get the insurance in your name. Oh, get a dashcam too!
if you are serious about staying longerterm, I would recommend asking an agent what your viable options are.
As already confirmed, you can extend your current 30 day exemption for another 30 days at immigration for 1900baht, I think as early as a week before your current stamp expires and you will get 30 days beyond the stamp date regardless.
Border bounce is an option, but they don’t like to see too many of those, it raises suspicion you are trying to game the system.
You can apply for a sixty day tourist visa in Singapore or Hong Kong etc too