My Name is on my Child’s Birth Certificate But I’m not the Biological Father, What are my Rights?

If you are not the father then it is not 'my child's birth certificate'

Some states say if you sign the birth certificate you are responsible.
I would check into seeing if you could prove through the family court that you are not the father with a paternity test if they would be willing to have your name removed.

If you are wanting to be a dad you go for joint custody and visitation