Unless he agrees, probably not, and even if he did, unless your new husband adopts them, your ex will still be on the hook for ongoing support (and will always owe the arrearages - that can't be fixed).
We can be reached at 507.334.0155. Our web address is: www. corbin-law-office.com This answer is not to be considered a response to a specific legal issue in a specific jurisdiction - it is to be considered only a general response to a hypothetical scenario posed by the questioner. For specific legal advice, please consult with a licensed attorney in your jurisdiction.
If the father consents to an adoption, you can have his rights terminated through the adoption. If he will not consent to the adoption, you can attempt to have his rights terminated. You must prove abandonment on the father's part or at least one of the other statutory grounds for termination, and the court must find that the termination is in your daughters' best interest. You should consult with an attorney in your county about the possibilities.
You need to consult with an attorney in your area to see what local judges will do in this situation. In N.C., one way to terminate a biological parents parental rights is to prove that it is in the best interest of the child and that the biological parent has abandoned the child and failed to pay child support as ordered. The court can consider the relationship between the child and the parent, the age of the child, and any relationship the child may have with a person seeking to adopt the child.
Sign up to receive a 3-part series of useful information and advice about child custody law.