The sides are open and because we're on an incline the water runs through when it rains.
If this is the case then no amount of dry bedding is going to help.
You need to fix the drainage problem by using grass swales, French drains, and/or diversion ditches to move the water around the chicken area instead of through it. In extreme cases you might have to hire heavy equipment to re-grade the area so that everything slopes away from the chicken facilities.
As a temporary aid you can set pallets and straw bales on the ground to give the chickens something to stand on that isn't soggy.
