Name Change on UK passport and Permanent Green Card

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I am trying to figure out how to go through the process of changing my name from my current name (First Middle Maiden, e.g. Anna Sarah Lee) to my married name (First Maiden Husband'sLast, e.g. Anna Lee Smith) on my UK passport AND my green card.

I have a conditional Green Card in my maiden name and have not yet taken any steps to change my name legally. My UK passport needs to be renewed before the summer and I will be filing my I-751 to remove conditional status on my green card this September. 

 

- Can I change my name on my UK passport application (which I know I can do with my marriage certificate) BEFORE I change my name on my green card? I will be traveling to the UK this summer before I can file my I-751, and when I return to the US I want to make sure I can get in even though my green card will have my maiden name still and my UK passport will have my married name.

 

- I understand that I will need to change my name on my green card before I can change it legally on my social security card. Is this correct? I have wanted to go ahead and change my name on my driver's license, but I need the name changed on my SS card first, and I believe I have to wait until I get it changed on my GC first. So complicated!

 

- I understand that I can file my I-751 in my married name if I send my marriage certificate with the application. Is this correct? I have seen posts by people who have said they have done this successfully, but I just want to be sure!

 

Thank you for any assistance you can give!

- Anna