Note: For simplicity, these answers address the situation of a stepfather who wants his wife's child from a prior relationship.
The law is the same if it is a stepmother, rather than a stepfather, who
wants to adopt.
FAQ #1 - What about the other parent?
If you want your husband to adopt your child by a prior relationship,
the rights of the child's biological father need to be terminated. This
can be done voluntarily if he consents (cheaper and easier) or, under
some circumstances, involuntarily, if he has failed to meet his
parental obligations. Sometimes, having a third party like an attorney
approach the biological father to discuss the possibility of an adoption
works better than trying to get him to sign by yourself, especially if
you do not have a good working relationship with him. That is what I
do, and if you want to talk over your situation with me, initial
consultations are always free.
FAQ #2 - What about child support?
When a stepfather adopts his wife's child, the rights and obligations of the biological father end. The biological father's obligation to pay current support ends when the decree is entered. If the Office of Support Enforcement is collecting child support, they need to be notified of the entry of the decree so that they will stop collecting current support. Notifying Support Enforcement is part of what I do when I handle a stepparent adoption.
Unlike current support, back (or past due) support is not automatically affected by a stepparent adoption, so if Support Enforcement has been collecting support, they are likely to continue collecting back support even after the entry of the adoption decree. Without a current support obligation, however, the back support can be paid off much more quickly than before. Also, sometimes parents can enter into agreements about the collection of back support, as long as the back support is owed to the parent and not the state.
It is also worth noting that a stepfather who adopts his stepchild takes over financial responsibility for that child, and so he will be responsible for child support if he and the child's mother ever divorce, just like he would be responsible for the support of a biological child.