Will a past DUI and dangerous driving charge affect my non-B visa renewal in Thailand?

First off you can't ever renew a visa :O You either get a new yearly extension at the immigration office inside the country or you go to a thai consulate in another country and you buy a new visa.

There have NOT been ANY new rules governing how S/A passport holders get extensions of stays INSIDE the country. The requirements for them are the same as everyone else.

There have been a few consulates in other countries that will NOT issue visas to S/A passport holders BUT that's a policy that's implemented consulate by consulate and not uniform across S/E Asia.

I'd say you need to go talk to the immigration office where you get your extension of stays at.. OR if you usually buy visas from a thai consulate I'd say check their website for the requirements concerning S/A passport holders.