Just wondering if there could be a place on your property that drains better? This is obviously not the best time to make a move, but it would be worth considering it in the future as is the cistern, mini drain field idea. Sounds like you'd have to get a whole lot of rock down before your bog improved. As others have said, the whole straw and leaf idea has it's drawbacks but will definitely do until you figure out what you are going to do. If this is your choice, definitely add enough so that water drains through it. The chicken poop smell is so much worse on wet ground, leaves, straw! I would also like to add that wet leaves are pretty slippery when wet too. 
We are fortunate that our yard is sloped gently, and the area we picked to place our coop and run is at the top of the slope. We added sand to our run on top of the existing soil, without a rock or gravel base. I put a mound of paver base (fine gravel and sand) around the outside perimeter. After a few months we added more sand to the run and when it got toward the rainy fall season we put a tarp over the run. Right now there is a combination of the wood shavings that fall out of the coop, the sand, some peat moss, and DE. It is very dry and the sand is very loose except for the inside one foot around the edges. Spreading some scratch and/or BOSS will keep the chickens moving around whatever you put in there.
Good luck!