Can you read the info I'm sending you in the link please?
"REGISTERING YOUR MARRIAGE IN THE USA: You cannot register your marriage through an embassy or consulate as marriage is a state-specific responsibility. Most states recognize the validity of legal marriages registered in Thailand. If you would like to register your marriage in the United States, you must contact the Attorney General in your state of residence to inquire about their specific documentary requirements. Some states may require your Thai marriage certificate to be “certified.” To certify a Thai marriage certificate, please have it translated into English and take both documents to the Thai Ministry of Foreign Affairs (MFA) Legalization Division. Then make an appointment and bring the MFA-certified documents to our office; we will authenticate the signature of the MFA officer by appointment for the appropriate fee."
They might have delegated certain services to different places so it's may be useful to navigate the website. In They might not have a single full service office in some places.
In the procedure in the picture it alliws you too choose location, that that might give you the information they person needs to do what they need to do.
Can't believe that we're helping Thai navigate thai website in English ...