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.