Spouse signing birth certificate

So I live in a state where you have to be married in order for your same sex spouse to sign the birth certificate. We’re currently engaged but have our wedding next year. We thought about getting married at the court house before the babies arrival the thing is my health insurance premium would go WAYYYY up for the cost of delivery. Currently it’s covered 100%. We talked about possibly just waiting till the wedding so we wouldn’t have to pay for the birth. That would then lead us having to go through an adoption process for her. Anyone had to do that? What was the process like as well as the cost? Thanks in advance.