I was wondering is there a law where the biological father can not sign off his rights unless the biological mother is married even if the person she is with wants to adopt the child?

You cannot "sign off" your legal rights - by that, do you mean does the non-custodial parent no longer have to pay support?

If that is the question the answer is that support HAS to be paid IF ORDERED until the child is adopted.

What the husband of the mother wants or doesn't want doesn't matter - the mother and father are the only two people who can decide whether the child can be adopted by the husband.

Is that your question?

(Also, a parent can simply not see the child; however, that parent is still responsible if support is ordered.)

If you had checked the Family Law forum (where this has been moved to) you could have learned a lot about signing over rights.

The bottomline here is that courts are very reluctant to terminate parental rights. Generally they will only do so to clear the way for an adoption or if the parent is a danger to the child.

Your location will often make the difference. Normally in "most" areas the mother will have to be married, for the new husband to file a motion to adopt the child.
The bio father normally can not sign over his rights unless the new step father can adopt.

In some areas long term relationships may be allowed

And I moved this to family law