I love this description.I wouldn't use the pellets at all. When wet don't they breakdown into a sawdust material. That is going to form a solid mat in the litter, not allowing air to reach down. It will promote the bad kind of mold and anaerobic decomposition that leads to odor.
Don't think about drying out the ground or trying to absorb all that wetness. Think about creating a lofty, multi-texured layer that sits above the wetness and elevates the birds off the wet ground.
OP might still want to dig some swales/trenches to keep any additional runoff from going into the run.
My daughter lives in NC, the rain has been ridiculously unrelenting.