My ol' 76 scamp
does the same. I noticed it when a pillow I stuffed into the closet would not come out on a hot day. It had stuck to the green inside closet walls. At first I thought of the fiberglass threads may have cought, but that was not so upon closer inspection. Next thought is that Fiberglass resins soften in heat, like epoxey glues do. Maybe the tackiness of that is what was grabbing my pillow.
Anyway, instead of moving to Alaska as the solution, I decided to put shelves in with rubbermaid square containers with lids on each shelf. All my stuff goes into containers and they sorta act like drawers with dust-proof lids.
I sorta like your contact paper idea except that the inside of closet is not very smooth and contact paper would probably be irregular and lumpy.
I want to solve it too, so I'm going to try coating it with polyurathane finish leftover from my cabinet woodwork.
WIll keep you posted, but it has dropped lower on my priority list.