No. He can pay for it but cannot initiate it. Adoption is a petition to the court by the person wanting to adopt. The exact process is delineated by state law, and varies from state to state. Normally your husband would be the petitioner. Step parent adoptions are straightforward if the biological father consents.
Please note that I do not practice in your state and that the above is not intended as legal advice, it is for educational purposes only. You should consult an attorney in your state to obtain specific advice related to your situation and to state law.