you do not have to change your name with the SSO first.
you marriage certificate is a proof of name change
on your AOS forms in the "Your Current Legal Name" you can put your married name
and put your maiden name in the "Other Names Used Since Birth" part.
this goes for any of the forms (I-485, I-765, I-131)
do you know what your SSN is? cause if you already applied for one you will need it before you can submit the forms.
if you haven't gotten the card back and it has been at least 2 weeks, you might be able to go into the SSO and see if they can tell if to you verbally, that is what we did.