The adoption process is fairly straightforward for stepparent adoptions, aside from the issue of getting consent. And, that seems to be no problem since you say the biological father is willing to sign. Parts of the ordinary adoption process may be shortened or waived altogether, such as the home study and the traditional waiting period. In some states you may need to be married before you may start the process but you are already. You can get the forms at your local courthouse in the county you plan to adopt. You'll need to demonstrate that you and your new husband are willing and able to take on financial and legal responsibility for the child. Remember that adopting a stepchild will end the other biological father's responsibilities to the children, including child support payments.
Also, your new husband may need to go through criminal background checks to satisfy the judge that he is fit to be a new parent. If it were me, I'd want to get an experienced adoption attorney to make sure your adoption is complete and legal. When the adoption is final, you can apply for a birth certificate with corrected information. You can try it yourself, though, too.