I don't think I would worry too much about wet hay; when Baby had her laminitis, the vet told me to soak the hay to leech out the sugars (a messy, royal pain in the buttocks.)
...as long as you don't have wet hay that starts molding.
Our goobers get their hay where I decide to put it; if they don't eat, their fat butts must not be too hungry.
I understand your distress about them being outside. Nicky, our old blind guy, hangs out against the barn wall--right where everyone driving by on the road can see him standing there. In addition to being somewhat embarrassing (as if I don't take care of our horses) I worried about him and used to spend a lot of time steering him inside, again and again, only to see him "pop" back out. I don't know why Nicky does what he does, but I had noticed a couple horses down the way (who only have a lean-to) standing outside with their heads snuggled up to the shelter, in the same way NIcky does.
We have a metal-roof on our barn; when it rains, it's deafening. The vet said Nicky does have some vision (light response); I'm dubious. I think he primarily steers himself by sound; and when it rains on that metal roof, he has lost his way to navigate. That's why he stands outside. I do grab his neck and take him inside when I feed him. Sometimes he stays inside; sometimes not. And the husband built him a "horsey-porch" over his stall exit; Nicky seems to like hanging out there.