I did an area about 50 x 70 and I’m in your region, so my experience might be useful.
Please use the heavy duty stuff! Landscaping cloth won’t cut it. I bought mine through a company in Vancouver, WA who makes it: CSI Geosynthetics & Erosion Control, 360-699-1426. Easy to find, I picked mine up at their warehouse.
Tell them what you are using it for, and they’ll recommend which kind/strength to use. I used one layer. One roll (about $290, four years ago) did my whole area.
I did not scrape away the soil, but my paddock slopes slightly, so it drains well. I simply rolled it out on a nice summer day, overlapped the edges and had 6 inches of screenings poured over it. You could do a layer of 3/4 minus then add screenings or pea gravel over it. I dislike pea gravel as it NEVER packs, ever. You MUST clean the manure off the paddock, as it will clog the cloth, and limit its permeability.
Feel free to PM with questions.