Hi! I'm a newbie to the chicken keeping world and wouldn't feel qualified to comment on this thread except for the fact that this morning I took a class in chicken coop building given at a local feed store. Deep litter was the method method demonstrated, although we were standing in slippery wet clay, the birds in the covered 8x12 coop were not. They were happily scratching away at least a foot above grade. There was absolutely no odor, although it hadn't been "cleaned" in a year. Once a year it is shoveled out and put on the vegetable garden.
If I was in the OP's situation, I would do as RonP suggested and put down wood chips. Coarse ones from a tree service at least 6 to 8 inches deep. Deep litter means as deep as it takes to maintain proper moisture and odor control. You might need higher belly boards around your coop. Those coarse chips won't become soggy right away like smaller shavings, so a good deep base of those and you can add whatever you can get whenever you need to add more dry browns. RonP is right about the chips also binding up excess nitrogen. In addition, the mycelia colonizing decomposing woodchips consume bacteria and even toxic chemicals.
We deal with high water tables here in the Pacific Northwest too, and putting down pavers might actually hamper your drainage, making your problem worse. So call your local tree services, follow a tree truck, bribe the crew if you have to. It sounds like you're needing to do something pretty quick and I wish you luck!
