Although a married woman can change her name simply by using her husband's last name and showing the marriage certificate where necessary, it's not so simple for children. To officially change a child's last name, a court process is required, and the court must be shown that both parents are consenting to the name change (or that the other parent is deceased or had their parental rights terminated).
That court process would have resulted in a document showing that your name was officially changed from one thing to another. These days she would not have been able to change her child's name at the Social Security office without that court document. But 20-30 years ago they were much more lax.
Without that court document, that last name she gave you isn't officially recognized by USCIS, so a replacement green card will be issued only in the same old name you originally had when your GC was issued, or your new married name.
Try to organize the following:
1. Your birth certificate. Get the help of your country's embassy or family members in your country to obtain a newly issued one.
2. A copy of your green card, or at least the A-number (this would be on various documents from your green card process, and may have also been written in your passport).
3. Any old picture ID with your original name, like an expired passport or state ID.
4. Your mother's marriage certificate (to show where the last name you're using now came from). If you can't find one, you can obtain a copy of it (or some other certified record of the marriage) from the state or county where she got married, because marriages are public records in the US.
5. Your own marriage certificate.
File form I-90 to apply for a replacement card with your new married name. Include a copy of all of the above.
For question 1 "current legal name", write your new married name as you want to see it on the card.
For question 2 "has your name legally changed" answer YES.
For question 3 "name as written on the green card" write the name as you see it on the copy of your green card, or if you don't have a copy you should write your name as you remembered what it was on the GC (as you said above, the name wasn't changed on your GC).
Attach a cover letter explaining that your mother changed your name from ___ to ____ when you were a child (give an approximate date if you remember), and you have been using that name ever since, but you have no court document of that name change and you don't know if she used a court process. And state that now you're changing your name to _____ now that you're married to _______.
If you have copies of both your old and new Social Security cards, that would be useful because both would have the same number so that helps with proving your name change.
When you go for fingerprinting, bring your marriage certificate along with your ID, because you're applying for a replacement green card with your married name, and your current ID doesn't have your married name on it. But only show the marriage certificate if they say something about the name discrepancy.
Which last name is on your marriage certificate? Your stepfather's last name or your mother's maiden name?
