Trying that from this moment forth, Noellereagan! Fingers crossed!
Just came back from the coop. Went over every bit I could reach (I'm short) with a long-handled brush -- inside and out of the coop and the run. Took a stick to the parts I couldn't. It now looks like some blizzaro sort of Halloween cotton candy! ....but I left the cobwebs intact on it because it grabs onto them much better with that sticky, dusty surface.
Took a few go-rounds since there were things I couldn't see from the inside and things I couldn't see from the outside. But I'm pretty sure now that I got it all.
My plan is to give a quick go-round and touch up every 2 days for 2 weeks and then watch what happens.
Thanks for the tip, Noellereagan!
I continue to marvel at the amount of information that's available around here!