I had another thought for you, Hip Hillbilly Farm. You might also want to check your overhangs. For instance, I use a tarp staked over my greenhouse frame (large hoop coop) as shown. We get moisture coming in the ends of the coop a few inches, but the run is so large, it's immaterial. The rest stays extremely dry. I put wood chips in that are damp from ground contact and partially composted- they've been sitting in my driveway for months. I get so much cross breeze and through breeze that any wood chip moisture quickly goes away, and I haven't had any respiratory issues. The overhangs keep it a lot dryer than laying the tarp flat up against the hoop, and make the rain travel farther across the ground before it can get into the coop, also avoids ground splashing except at the open ends.
I'm in north Alabama USA, so we get major humidity and hot temperatures most of the year, and quite a bit of rain usually.
That's a beautiful setup for the Steamy Southeast!