An adoption by your fiance would be considered a "second parent adoption" rather than a "stepparent adoption", but the process is very similar, especially with regard to the biological father.
Your fiance would need to file a petition to adopt your son. You would need to join in the petition as well.
As it sounds like you know, you need to deal with the rights of your son's biological father. The best way to do this is to get his consent, on a consent form that meets the requirements of the law. An adoption lawyer can prepare this form for you and, if necessary, meet with the child's father to request that he sign it. You can check the American Academy of Adoption Attorneys website (link posted below) to find an experienced adoption attorney in your area. You can also find a list of Washington adoption attorneys on pages 12 and 13 of the "Adoption Information Exchange" resource list found online at the link posted below.
If it is not possible to get the father's consent, there are other procedures you may be able to follow to terminate his rights so that the adoption can go forward. The process gets somewhat more complex at this point and it would be best to have an adoption lawyer helping and advising you about the options, considering the specific circumstances of your case.
Another requirement of the process is to have a report done by a court-appointed adoption social worker, who will tell the court whether or not the proposed adoption will be in the best interests of your son.
If the social worker's report is favorable, then 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 fiance the legal father of your son. As a part of the process, the biological father's obligation to support your son will terminate, and your fiance's obligation to do so will begin. The biological father's past due support obligation will not be affected by the adoption unless you enter into an agreement with him to release him from the obligation to pay past due support.
After the Decree of Adoption is entered, application can be made to have your son's birth certificate re-issued to show your fiance as his father.
I hope this is helpful. Best wishes.