Do I need to do a visa run after getting a new passport with my non-immigrant B visa in Thailand?

Travel with both passports, that's it.

When you get your new passport simply take it, your old passport, and the letter from the US State department you got with your new passport to immigration. Having a copy of every used page in your old passport can save you needing to go make copies of pages. This is free to do. You don't leave the country. Immigration first puts a database reference number stamp in your new passport, and transfers your entry stamp into it. It is a very easy process.

Your visa was used when you entered Thailand. There's no visa to move. You can get your entry stamp moved to your new passport at your immigration office.