The photo helps a lot, not as bad as I imagined. If that ditch dirt is not all clay then sure but mix in some sand. Also yes, let the chickens have the grass and such before you cover it. Then you won't have organics rotting under your soil/sand base adding and holding moisture.So what you are saying is I'd build a ditch a few feet back from the coop and run around the entire area of the coop/run and use the dirt to raise the level of the coop/run area, and then fill the ditches with rocks? I could run the ditch down to my berm area, but it doesn't really go anywhere after that, the water pretty much sits in the berm ditch until it absorbs through the ground in a few days (after a large storm). Trying to picture this in my head and having some trouble.
I imagine your ditch as a DIY version of what we call in construction "drain tiles" or "French drains" but I don't think you need any plastic tubes unless you don't want to capture the roof water, then gutter it out to your ditch with it.
Otherwise go to a landscape supply yard (not a big box store) and show them the photo. They'll sell you the right kind of rocks for drainage and probably have a sandy soil mix also.
There are entire books on soil. For example first heave it's an issue in sand because the sand doesn't expand when it gets wet. The landscape places have read these books and sell their wares cheaper.