I normally leave the north and west walls solid due to prevailing winds. However, the last two buildings I built have close to 3' X 4' open windows on both east and west walls and the wind blows right through at roost height. Sometimes snow blowing in. Chickens have been in there down to -19F...
Cut big holes in the walls. Disease spreads most rapidly in the warm moist environment of a closed up coop with little to no air exchange. Big exchange of oxygen it the way to keep them healthy.
Just because you are cold doesn't mean your chickens are. Their reluctance to go out into the snow...
Are you sure they aren't walking over the hay to the water fount at any time during the day? Keep in mind that they drink a lot less water in cold weather and they will also eat snow for hydration.