Before you can proceed with a step-parent adoption, you must first terminate the biological father's rights. From the information you provided, it sounds like you would be doing so on the grounds of abandonment (generally, this requires failure to maintain contact and provide support for a period of at least 6 months). The termination process is separate from the adoption, and although you can file your adoption petition while the termination is pending, the termination must be completed before the adoption hearing can take place. Terminating the father's rights does not relieve him from the arrears he already owes.
If you are able to locate the father, he might be willing to sign a consent to allow your husband do adopt your daughter. The consent, when drafted appropriately, is irrevocable, so the father could not later change his mind. If he signs a consent, it is not necessary to proceed with a separate termination proceeding.
If you are not able to locate the father after diligent efforts, the court might grant you permission to give him notice of the proceedings by publishing in the newspaper. In other words, there are still ways to proceed with the case even when the father cannot be located.
There are other steps you must take in both the termination and adoption process, and it would be best to speak with an attorney who can answer all of your questions and present your options.
** This is intended to provide general information only and does not constitute legal advice or otherwise create an attorney-client relationship **
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