I know and lived with my birth mother for over 17 years. She has been deceased since August, 1975. I lived with another family and recently found they adopted me at the age of 18, after misplacing my original birth certificate and requesting a copy. I have only used my birth name on all credentials. Since the new birth certificate reflected the last name, etc. of the family noted above, I have not been able to renew my license, as the new birth certificate does not reflect the same birth name or show that I am the same person.
It is a great step to proactively resolve your issue - otherwise you may have issues with identity theft, credit issues etc.
I have seen this issue arise in a variety of situations. You will need the assistance of a good attorney in NJ. A court can issue a declaratory judgment, simply explaining that you had one name between the years of birth and adoption (no need to mention adoption) and basically, Jane Doe also known as Sue Smith are one and the same. The court probably won't go so far as directing vital records to issue a different birth certificate, but at least you will have the issue of two names resolved.