Go here:
Birth, Death, Marriage & Divorce Records - New York State Department of Health and follow the links to download the application forms for the records. Unless you have reason to get certified copies for your records, check to see if they have uncertified or genealogical copies because they may be cheaper.
Her Social Security card was issued in New York, so there's a good chance that's the only state she ever lived in. I recommend you apply for all records - birth, first marriage, divorce, second marriage and death - in NY and see how much they have, and what clues the records hold. And if there's no marriage besides the one to your father then there's a possibility the twins had the Stranieri surname. Report back to us your findings and we'll go from there.
While they won't give you the twins' birth certificates, they might have a birth index available where you could get their surnames and birthdates by doing a search on your mother's maiden name. You could call and ask if an index is available. You might not be able to get her first marriage and divorce record without knowing the husband's name or if he's living, but there is a possibility that the information of the first marriage/divorce will be on the second marriage record. (states vary in how they keep records so I can only give you generalities on these matters)
The SS death record is the only record I've found on her, as NY does not make the records above available to the general public. I did not find an obit for her, but there is another searcher here who has access to other obit archives and she may find one. If there was a funeral, perhaps you know of the church? That would be another place to check. From the surname I would guess she may have been Catholic. The church would have a record of her baptism and possibly confirmation on file. They may have yours and the twins' records, too. Besides listing the parents, they would list the godparents and confirmation sponsors - which are often relatives. You could also check with the children's home where you were placed. They may have record of your siblings in your file.
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