What is the process for renewing a non-immigrant O visa (marriage) and moving to another province in Thailand?

Remember, when you apply for a year extension based on marriage to a Thai national you get a 30-day under consideration stamp and you have to go back to that immigration office 30 days later to get the year extension inked in.

And there's a good chance that they could do a home visit during that under consideration. As well.

So I wouldn't pick up and move provinces until you get the year extension inked in (or of you do move know you have to go back to get that year inked in when the u/c stamp runs down)

Then once you do move all you file is a TM 30 and then start doing your 90-day reporting at the office that serves the area where you live and then the following year apply for your extension through them.