What can I do about damp straw in my run?

I have a covered run attached on one side to a storage building with a concrete floor on the uphill side of the run. It is also wrapped in heavy plastic for the winter. There is about 8" of straw in the run. I was shocked to see at least half of the straw damp this morning and this evening it was almost completely damp. We have water standing in the yard from snow melt and recent rain so I am assuming the dampness is somehow slowly seeping into the straw from the wet ground - water is not running in. Doesn't make sense because it would be seeping uphill. I think I need to do something about the damp straw quick! I was just reading another post where wood chips were suggested. Can someone tell me what size chips should be used. We get aged wood mulch from a local sawmill that is fairly fine and we use it in our garden and flower beds as mulch - the chicks dearly love to dig in my beds! We can get larger chips from the same place which looks to be about 1-2" wide and thick and 4-6" long. If I go with the wood chips, which size should be used? The larger chips don't sound too comfortable to walk on but then I'm not a chicken and a newby at raising chicks at that. Also, I'm concerned that the chicks might get splinters from the wood. Any advice on what I can do will be greatly appreciated. Thank you.