We are having the same problem here in south jersey right now also. Lots of rain and (unseasonable) humid weather. My poor chickens feet are completely covered when they walk. And the smell- OH THE SMELL! After a lot of researching, I put loads and loads of a mixture of pine needles, dirt, gravel, leaves, composted pine shavings (from their coop bedding that I clean every 2 days and dump in a pile out back). Took me 2 hours. I spread it all throughout the pen, paying more attention to the problem areas that tend to puddle. The smell instantly went away and now they have a soft floor to walk on and their feet are not standing in mud. We are supposed to get more rain today and tomorrow also so I am wondering how long it will last. But at least now I have something that is compostable. If I need to add more throughout the week, I'll just add it on top of what is in there now. It will eventually compost. I thought of doing hay, but it's $7 a bale here and I didn't want to have to rake it all out in a few days. So, I am really hoping this works. (fingers crossed)
**Update**
Well it did rain pretty much all night last night and much to my surprise, there still is no sign of mud in the run. And the smell is still gone. When they walk, they don't sink either and mud is not coming to the surface. I think I finally found a way to keep the mud at bay and for FREE, so happy!