This is a sadly common scenario. I suspect WA law is not much different than GA on this; and if so, he can go through the motions of re-establishing contact with your daughter, and doing similar things with payment of child support, and the law, and judges, are going to let him preserve his parental rights and deny your current husband's petition to adopt her.
Maybe a WA lawyer could advise otherwise, but GA law is pretty forgiving of parenthood that most of us would call "bad," if not "unfit."
If you have a good rapport with your ex's mother, or someone like that whom he trusts, and you can get that person on board with the idea of your current husband adopting, you might be able to get him to agree with it. (Of course, his mother would probably need to trust that you would still let *her* see her biological grandchild.)
Then there's the issue of the back child support. In GA, and in most states, I believe, that doesn't go away after an adoption is granted. In fact, it's actually illegal to promise to forgive back child support if the man will consent to an adoption (though that surely happens all that time, against the advice of GA attorneys). But there's no argument that *future* child support obligations stop once an adoption is granted --- and sometimes *that's* enough to persuade a man to consent to an adoption.
So while it's a tough situation, you've got some options.
You file a petition and serve a petition for adoption. the first step is to terminate parental rights of the father. make sure you get proper service. if the ex fights, then you have to go to trial and, honestly, your chances of prevailing aren't that good. if he doesn't fight, then you an order terminating parental rights, do a home study / placement report, then you go back to court one more time wiht all the paperwork.
here is the sweetener for the ex: if his rights are terminated, he is no longer liable for future child support, though he remains liable for past child support.
It is possible that you could resolve this in a pretty straightforward, low-conflict way. If your daughter's biological father will consent to having your husband adopt her, a stepparent adoption in Washington can be pretty simple.
In Washington, we have a law that might help you get the biological father to consent to the adoption. This law provides for enforceable Open Adoption Agreements. If you and he can agree on some level of communication and contact (letters, photos, an occasional visit, whatever...), you can both sign an agreement saying that he can stay in touch with the child in whatever way you have agreed upon. Then that agreement can be presented to the judge as part of the stepparent adoption proceeding. If the judge signs it, the biological father has a court order assuring him that even though his parental rights have been terminated by the adoption, he won't have to completely lose touch. This is really the best of both worlds for him - the adoption terminates his obligation to pay future child support, but the Open Adoption Agreement reassures him that he can stay in touch.
As far as the basic procedures of a stepparent adoption in Washington go, your husband would need to file a petition to adopt your daughter. You would need to join in the petition. Hopefully you will have the consent of the biological father. It is not impossible to go forward without his consent, but it is more complex and more expensive, and if he has been paying support recently, it will be difficult to convince a court to terminate his rights over his objection.
You also need to have a report done by an adoption social worker, who will tell the court whether or not the step-parent adoption will be in the best interests of your daughter.
After all of the necessary documents are filed, a date for a court hearing is scheduled. If the court approves, the judge signs a Decree of Adoption that makes your husband the legal father of your daughter. The Decree can also change your daughter's name if you wish.
After you have a Decree of Adoption you can complete some forms and pay a fee to have your daughter's birth certificate re-issued to show your husband as her father, and showing her new name if her name was changed by the decree.