I agree with most comments, I have a Thai partner and 2 children, I would never risk anything that could stop me from returning to them. Even though your over stay is unintentional, Play it safe and abide by there laws.
Don't listen to the Don't need one crap, you have an accident technically in this country you are unlicensed, I was pulled over today when they block the whole road, I handed over my Australian licence my IDP and my passport, he said where is my thai licence, I said your holding my IDP, he apologised and off I went, probably looking for a pay day or he couldn't read English, but yeah definitely have a licence that's legal to drive here, to cover any serious issue's