You run a very real risk of letting human bias lend more weight to some information that supports your beliefs and less weight to those that do not, but ok.
In most states, to change non-trivial information on a birth certificate takes a court order. See if you can find that information at the court house. It sounds like you have the key date.
Look in old newspaper records to see if your birth was announced. Some papers announce all the local births, maybe with the parent names.
Ancestry.com can look up birth records although they might charge you and they might have nothing more than what is already on your birth certificate.