I started with the SSA (avoid the one in Brentwood if possible). However, they used the marriage certificate to confirm the name change--not sure how they would do that for you without having your new name on the marriage certificate. I was told to put my new last name on the marriage certificate and did so.

My advice would be to call SSA (expect long wait times) and ask if they are able change your name based on the marriage certificate--if they can, that would save you time/money. Otherwise, you may have to go to the courts.