This is probably going to be unpopular, and this is my first summer as a chicken keeper I'm by no means an expert. I disagree with the posts that say the fan needs to go. I see you live in Kentucky. I live in Texas and it's 113 degrees (heat index) today with 80% humidity. My poor birds were miserable in their coop despite having excellent cross-ventilation...until I installed the fan.
When it's this humid with no wind, it doesn't matter how many windows you open, the air doesn't move. Misters aren't an option in the run, and any evaporative cooling just perpetuates the moisture problem.
I have a fan that looks almost just like yours. A 10" galvanized exhaust fan. It's mounted in my coop under the covered run, and what we did to help stave off buildup is:
1. selected a low-dust coop bedding - hemp; and
2. reverse-installed the fan.
It's mounted with the intake outside under the run so that it's not pulling air from inside the coop and exhausting it out. It's pulling air from outside, under the run, and pushing it into the coop. This makes for less dander buildup because it's harder for dander to make it past the moving air. It also creates a breeze in the run and keeps air moving in the coop. With the windows open, I've been able to keep the coop around 90 in the day and in 70's - 80's at night.