Renewing your passport with your marriage certificate to change you name only changes your passport.
You'll still need to change your social security card, US driving license, name on your US bank accounts, etc and some of those need to be done in person.
If you've got a BRP rather than a sticker in your passport, you'll need to change that within 3 months of changing your name on your passport.
But there's nothing saying you can't use your married name for everything but keep your passport and visa in your maiden name.
As long as your passport and BRP (if you have one) match, you're fine.