I don't know how available it is in other areas of the country...but if you can get it, the absolute, hands down, no question, BEST litter for a run is...
Recycled dirt from marijuana growers.
I do not grow or smoke the stuff, but I live in an area that sees several "free dirt recycling" posts a week on Craig's list. Since the plants get the very best soil to start, and then are dried in the soil, after harvest the soil is fluffy, light and completely bone dry. I have poured gallons of water directly into a trash can full of the stuff to try and get it ready for use in my garden beds, but the water just evaporates and/or runs off. After processing by the growers, and from being under the grow lights, it can no longer hold moisture without first being aged by earthworms and such.
I put recycled growing soil in my chicken run, duck run and turkey run and the moisture in the droppings dries as soon as it hits the dirt. All I do is rake the runs once a month and I don't even have a fly problem in my runs. My yard where they free-range is consumed by flies, but the runs are clean and odor free.
The turkeys and the ducks sleep on the recycled dirt and again, no odor, no moisture. I just rake the top every couple of weeks (not replace, just stir) to fluff it up and redistribute from where they've burrowed in.
If I had known how good this stuff is, I would not have built my chicken coop with a breathable wire floor. I would have put in a solid floor so I could put in in the coop as well. The pine shavings that I have covering the wire floor in my coop get terribly stinky and compressed with poo and moisture and have to be changed often.
If you can find a source, you will LOVE the results. We tried it as a favor for a friend who needed a place to dump his used dirt and we will never use anything else.
We live in a part of the country that sees almost as much rain as Seattle and are right on the coast so we have precipitation in the form of rain or fog every day. Even with our wet weather, still no stink or moisture in the runs. I'm telling you TRY it.
Or you can buy some grow lights and try to dry out potting soil yourself. But why not recycle if you can?
Disclaimer...I am not advocating the growth or use of any illegal substance. In my state, however, it is legal with the proper licensing. And while my family does not use it, I respect others right to do so under the law. My only purpose with this post is to share my terrific results at keeping a clean, odor and fly free run.