Is the wetness because drainage issues? Is there any sort of cover or could you add even just partial rain cover? We have couple of sheets of clear corrugated plastic over the run to give girls dry'ish corner to escape the worst of the weather and still be outside rather than being stuck in shed all day long.
I put absolutely all sorts into our girls run...annual weeds, chopped straw, woodchippings, sand etc...and the girls have great time mixing it all together. Once the 'bedding' material level gets too high for the run (starts spilling over into garden after 8")..it gets emptied and wheeled into allotment and spread straight on beds or piled on somewhere to be used later on.
And then the fun start again...piling up the stuff...day by day, week by week...