Contact the hospital and they some guide you. You can change any mistakes that were made
We messed our first childs birth certificate up. The county advised us to contact the secretary state.(We're in CA) It was free to change as long as we did it in the first year of life.
After the birth certificate is registered, you should be dealing with the state to fix, like the office of vital records or something. I just requested a birth certificate from my state (VA) and they had a form to correct errors on the birth certificate.
Did they not give you the form at the hospital to fill out? It has your current legal name as well as your name when you were born.