Any of the moss that has turned black has already been killed. There's a few different formulations of Moss Out. The one for lawns contains ferric sulfate as the active ingredient. It can persist in the soil for quite some time and I don't think you can leech it out with water. Other Moss Out formulations use either ammoniated soap of fatty acids or zinc sulfate. The ammoniated soap of fatty acids is the most biodegradeable but I think what's done is done.
I had some moss in a part of my lawn a while ago. A number of factors contribute to moss growing including compaction, not enough sunlight, low pH, etc.
Send a soil sample in for analysis, not too much money, to get a better sense of the fertility and pH of the soil and make appropriate corrections.
To deal with the compaction you should core aerate. You can hire someone to do it, rent a machine and do it your self or you can buy a manual core aerator. I have a review and instructions for the Turf Hound Aerator on my site. It's what I used to deal with my compaction issues. It's good to have to hit trouble spots if you don't aerate every year.
After core aerating spread some compost over the area. About 1 cubic yard per 1,000 sq feet. Raising the organic matter of the compost will help alleviate compaction as well. If spreading around all the compost isn't appealing I've read and seen good results with regular compost tea sprays after aeration though don't have much experience with it myself.
Since it's shady, the soil won't dry out quickly after a rain. If you can keep people off of it during and a couple days after rain it would help.
After topdressing with compost overseed with a good shady lawn mix that will be primarily composed of fine fescues. Make sure it has some creeping red fescue in it because it's a spreading type grass unlike other fine fescues. I have some shade grass seed recommendations on my site as well. You're near Oregon where much seed is grown so you shouldn't have a hard time finding a good blend.
I believe there's still time in your area to overseed before it gets too hot so you can try it since fine fescues germinate fairly quick. If it comes in patchy overseed again in the fall.