Adjustment of Status - I-485

Hi everyone, I recently received my marriage based green card. All the Applications were sent with my maiden name. At the time of the interview, we requested that my last name be change to my husband‘s last name. My green card arrived with my husband‘s last name. Which is what I wanted. Now, I’m in the process of changing my name with other agencies. Will my green card be enough to use as proof of name change? Do I need to change my name with the court as well. My marriage certificate has my maiden name because until now I had not change my name. Can you all share your experience with this? Can I just take my new green card to get my new ss card? I have an old one that I used with my EAD while I had TPS. I know it will be the same number but they will probably have to remove that message that it is only allowed for work. Thanks and let me know!