What is the process for obtaining a marriage affidavit and passport certification as a US citizen at the US embassy in Bangkok?

You don't need your passport translated, just verified as being real. The notary service on the right to marry document is $50, or baht equivalent. The document has three boxes you can check, single never married, divorced, and widow. The last two need supporting documents, your divorce decree by the court, or in case of widowed the death certificate. The US embassy doesn't verify what you check. They only notarize that it is you signing. Keep that in mind if getting the other documents is too difficult. Get it translated, then go to the MFA to get the accuracy of the translation verified, and them stamping it.

Be aware that some translation services also offer a service to do the MFA part for you, at a fee. There are some translation services that will intentionally make a mistake so you have to come back and get it corrected before the MFA will verify and stamp it. They do this so you will use their service in the future, as it is a long way traveling back and forth to immigration on Pak Kret road.

Make sure you get name translated correctly as can affect future documents.

The MFA doesnt do the translation. You have to get them done elsewhere. I used one by old british embassy (express translations)

I needed passport done as well I think (been a while) but did use it as well afterwards to get my pink ID and yellow book