Hi Ludlow,
My husband did all the paperwork, so I don't know all details. He went to the local district office (Khet in Bangkok or Amphur upcountry) with the documents we had, and asked what is needed. We had to use an official translation company to translate my passport and the marriage certificate into Thai, I remember that. The translations then had to be verified by our embassy, and the certified translations had to be certified again by the Thai Ministry of Foreign Affairs, Consular Department. The translation office arranged for these two certifications for us, they do it as a routine service for a small fee as it is often required. I think this was all, but please double-check with the district office.
The district office then changed our status from "single" to "married", and we received a certificate for that. So, it is important to know that we did not get married again, they just acknowledged that we are married already, which was fast. In fact, it was immediate on the day when we brought all the documents and certified translations.
As for the visa, I am on a non-quota immigrant visa and am listed in the blue tabien baan. I don't know whether this makes a difference.