They do stay wet, until they drybut it seems like they would stay wet to me especially if deep.

I don't have them too deep, an inch or three, they don't need to be deep to 'eat up' the poops...more like slow cold compost.
If you are 'making garden soil' in the chicken run, best not to use wood at all, IMO, as it takes forever to fully break down and you don't want mixed into your garden soil as it will suck up the nitrogen present stealing it from the garden plants.