How does step-parent adoption work if no father is listed on the birth certificate?

My daughter is 6. My husband and I have been married for 2.5 years. He would like to formally adopt her, but we aren't sure what the process will look like since there is no father listed on her birth certificate. I have not had contact with the biological father since she was a few weeks old. He has verbally relinquished all rights (visitation, child support, etc.) and has never legally filed for any paternity establishment. Honestly, I don't know how to contact him if needed for termination of his rights.