"Also, does my current spouse need to adopt her in order for this to happen? "
The dad can't just "sign his rights over" but he can consent to termnaiton of his parent/child relationship AND adoption by your spouse.
Adoption will not affect past due support.
A biological parent can voluntarily surrender their parental rights only as long as they are being surrendered to someone else. In your situation, if your husband is intending to adopt your child, the father can voluntarily surrender his rights in the adoption proceedings. (The father cannot sign over his rights unless your husband is adopting.)
Once a parent's rights are terminated, that parent is no longer obligated to support the child. The parent is also no longer entitled to visitation or inheritance rights. If your husband adopts your child, and the father surrenders his rights to allow the adoption, the biological father will no longer be responsible for paying new child support assessments. However, this ONLY applies to new payments. He will continue to be liable for all past due arrears (all support that was owed up to the effective date of the adoption).
If you and your husband decide to proceed with a step-parent adoption, you should hire an attorney to assist you. While the process is not overly complicated, it is extremely specific. It is not advisable to attempt it on your own.
The above answer is a general explanation of legal rights and procedures. It does not constitute legal advice. Nor does it establish an attorney-client relationship between the individual posting the question and the attorney providing the answer.
There are 2 issues here: Child support and Parental rights. The Court is always going to ask: What is in the best interest of the child? Therefore, the biological father will always be responsible for Court-ordered child support. Yes, his license will be suspended if he is far enough in arrears. Further, he will be responsible for all of the child support he owes. He cannot simply pay you off and then abandon his rights.
The prior responses cover the signing over of parental rights. It is advisable to consult an attorney because there may be other relevant issues which may change the outcome.