I feel sorry for you, you just followed the stream of people terrified to be out of Thailand one day to another, this grace period was really obvious as all the country in the region gave one, Thailand was just the last to announce it.
so you would have been in amnesty anyway and you could have extended your stay for 60 days at the end of July in case the don't extend the amnesty, which I presume they will with all border closed and no commercial flights on sight...
it was a while ago as since 2019 in most of the land transport office you'll need a certificate of residence for get the DL, then I want to add that if you need to check in in my hotel showing a passport as a foreigner visitor is mandatory as we need to fill the TM30 form. So again saying that travel as a foreigner in Thailand without your passport I doable can be right, it is a good idea? Certainly not.
because a tourist is now allowed to have a thai driving license in Thailand... He's a TR he need passport and yes he won't have any problem with the permission of stay expired due to the amnesty...