Please note I am not licensed to practice in your state and this answer is in the generalist of possible terms.
It is possible for you to do all those things you would like to do thought a petition for a step-child adoption. In the word of an attorney's website from Tenn. "Step-parent Adoption is where a step-parent desires to adopt his/her step-child following marriage to the child's parent. The biological parent married to the step-parent joins in the petition for the purposes of consenting to the adoption. The rights of the other biological and/or legal parent must either be terminated or that parent must also consent to the adoption."
The procedures and rules are simplified for this type of adoption, but it is still a legal proceeding with a petition. The difference between a step-child adoption, and a regular adoption is usually in the lack of a home study / placement part of the process.
You will need to put the BioDad on notice of the adoption. There are different ways to do it, but in cases where you need to terminate the rights of another, it is highly recommended that you contact a local attorney with experience with adoption. You may not need full representation, but you will need guidance.