@Sunida *************
I think there is a definition mismatch. You were told "notary" or "notarized", but what I think you need is the full authentication/legalization process for your marriage certificate.
As Pom Argent mentioned in an earlier reply, the process is as:
Step 1. Obtain certified copy of the marriage certificate from the county where you were married in the US
Step 2. Have that authenticated by the Secretary of State in the US state where you were married.
Step 3. Have that authenticated by the US State Department in Washington DC. They will not do so if step 2 has not been done.
Step 4. Have that authenticated by the Thai Embassy or Consulate that serves your region in the US. They will not do so if step 3 has not been done.
Step 5. Get a certified translation into Thai of the entire document stack above.
Step 6. Have all that approved by the Ministry of Foreign Affairs in Thailand
Step 7. Take all of the above to a local Amphur in Thailand to have your family status updated to show that you two are married. They will give you the Kor Ror 22 form.
The US embassy in Thailand, as far as I know, does not offer such document authentication/legalization services, such as authenticating a marriage certificate that has been authenticated by a US state. Document authentication/legalization is different from notarization. I think that is where some of the confusion is coming from.