You should review your specific facts with your attorney to find out your legal options.
In WA, unless both legal parents are dead or are found by competent courts to be unfit parents, it would be difficult for a legal stranger to have court-ordered time with a child when the legal parents do not want the legal stranger to have time with the child. A "non-biological, non-adoptive parent" would be a legal stranger when there is a divorce from the child's legal parent.
Most children do not care about who is related to whom by biology or marriage. In an ideal world, the adults would consider the desire of children to maintain contact with other children and other persons with whom they have bonded.
WA has a statute that gives rights to those who have been in a parent-like realtionship with a child (de facto parents), consented to and fostered by the biological parent, of a duration long enough to establish bonding, if the person requesting rights has lived in the same household as the child and fulfilled parental duties without expectation of financial compensation. However, case law is pretty clear that stepparents cannot be considered de facto parents. Although you may feel like a parent, you probably cannot be granted the rights of a parent. If it is possible, and the biological parent is willing, you might look into adopting the child.