Updated today's SSA visit:
Just walked in my local SSA office without appointment. Handed them my NC, driver's license, Name change court order, and application. All originals. The officer asked if I changed my name because I got married. I said I just recently got my citizenship. On the oath ceremony, I change my last name to my husband's last name. It seems like she didn't understand the connection between getting a citizenship and name change. She asked for a marriage certificate. I said I married four years ago. I don't have it with me today. She started to work on the screen and checked my documents. I guess a court order suffices.
I though they were going to take away my NC until further notice. In the end, they returned all my documents and gave me a receipt. New card will be arrived in two weeks. I checked my online account later the day. My name has been updated, but still showed non-citizen. I'll check again in two weeks.