It sounds like you have a lot of clay in your soil. I have soil that does the same thing in one of my gardens.
One of the best materials for a chicken run (often called the "gold standard") is wood chips, like what you'd get from tree trimmers who chip up what they cut. (NOT like the wood chip mulch you can buy by the bag at garden supply stores; that's often dyed or chemically treated.)
See if you can find someone in your area who is getting trees taken down or trimmed. There are places online where you can look up sources for chips. The arborists don't want to have to pay to dump the chips in the landfill, so often you can get them for free.
A 3-4" layer of chips in your run will have your chickens scratching it up, happily. Some leaves and other garden debris will work too. It might be a few weeks before you can get some, but keep it in mind.