My long-former gf had given up a daughter to adoption and was trying to make contact again. She worked with a charity based in Oxford that used to help with matching, but as someone else said, it's long-defunct.
I think that all you can do with SS is ask them to leave a message on your/DM's file saying that you are looking for her and want to make contact. So if she does come looking, SS should advise her of that. Make sure to keep address, email, phone numbers up to date.
It's a long shot, but you could do a DNA test on Ancestry, because even if your DM doesn't, she may have had other children who also do DNA tests (as may your birth father). And even if you don't find any results immediately, the information is constantly updated and compared with every new test they add to the database.
Did you/can you seek any info from your adoptive parents? I think that many adoptive parents understand that sooner or later you are likely to want to know about your birth parents and there is often information/documents salted away somewhere.
If you feel that you are getting close to being able to make contact, tread very carefully and consider using an intermediary to make contact. Your DM gave you up (or had to give you up) and has moved on in her life, it can be a big shock.
Don't have expectations of a quick result, sometime it can take years. And sometimes the other side just doesn't want to know.