Go speak to immigration first, you will need Documentation in English and Thai, but each Office different, also Marriage certificate from the Amphor needs to be updated
Yes, with a Non-O visa, you can depart and re-enter Thailand, but you'll need a re-entry permit to avoid invalidating your visa. If you leave without a re-entry permit, your Non-O visa will be canceled
What about getting remarried in Thailand should that work, Have you spoken to the local ampour on what you need, would an translated certificate work. Good luck, it's not easy to open Bank account with out the correct paperwork.