I changed my name on my driver's license and social security card in the same two day period. I received a temporary driver's license and waited for the new SS card.
My first problem was other IDs, like going in to Sams Club and I didn't have "valid" or new ID with both my picture and name.
My worst problem was that the Social Security office misspelled my new name while the driver's license photo wasn't taken right and came out gray. When I went to the DMV for a new photo, they wanted valid ID - and the name on the driver's license didn't match the new SS card, and Social Security said I didn't have a valid driver's license to prove my identity because I didn't have a new one with my picture and new name on it.
I showed up with student IDs, membership cards and birth certificate and marriage license to get Social Security to recognize me as me and the name change and correct their spelling error. Then I was able to take that to get a new driver's license.
What should have been one visit to each office was half a dozen and months before it was all corrected.