Changing your name on official papers like Social Security or a drivers license, or passport, etc is not the same as a legal name change. Those are separate elements of the complete process.
First, you legally change your name.
Second, you get new SS card, and license etc, that reflect the new name, using the proof of legal name change. You also need to update bank accounts, etc.
It is not clear from your question whether your name was legally changed when you married or not since you say you need to change your name but then imply you plan to do it at the Social Security office. The Social Security office cannot change your name. That is a function of whatever state you live in. Social Security can only issue new papers that reflect your new name. The process and rules for a legal name change vary depending on your state. It is usually very easy to do when you get married, and somewhat more complicated (but still not to hard) to do it later. Most states have a question on the marriage license application asking if you want to change your name. If you answered yes then (depending on the state) then your name has probably already been changed. If not, the process often involves filing papers with a civil court and publishing an announcement in a local paper. This process should be done before you go to switch the names on official documents.
If you did already change your name when you were married, then you need to take the proof, which could be a marriage license (again depending on your state and particular circumstances) to DMV, and Social Security, State Department (passport) etc to get new papers.