It depends on where your run is located. If it is in a low spot where water runs in and stays, it's going to be wet no matter what you do when it rains. If it is positioned where it can drain, you have a lot better chance of it staying drier. It helps to keep the water out in the first place if you can. That’s not always possible, especially if you have a large run. Here is an article on fixing a muddy run that has some good information.
Pat’s Big Ol' Mud Page (fixing muddy runs):
https://www.backyardchickens.com/web/viewblog.php?id=1642-fix-a-muddy-run
If it can drain, sand is a great material to use. Water drains out of sand really really well. The thing is that the water needs some place to drain to. If you built the run in the bottom of a bowl, the water has no place to go. Chickens love scratching and dust bathing in sand. They can use sand as grit, the coarser the better. I suggest using a coarser sand like construction sand and not the really fine sand like play sand. It drains better, is better for grit, and does not stick to your clothes as bad.
A couple of possible disadvantages to sand. If your run is set up where a stream of water runs across it when you get a rain, it can wash away. You may need to build a barrier at the lower side of the run to contain the sand, or, even better, build a berm or swale on the upper side to keep the water out. But remember to still give it some places to drain.
The other possible disadvantage to sand is that, depending in the type of soil you have, it can work its way into the mud underneath over time. You may need to replace it. How often depends on your specific conditions, obviously. One way to help with this is to put a layer of gravel under it. I suggest smooth gravel so there is less chance of them cutting their feet when scratching.
Even if your run is not in the perfect spot, you might be able to build a barrier around the bottom of your run to contain the sand and build the sand up fairly deep so it can drain.
We have all kinds of different runs in different climates in different conditions and in different sizes with different amounts of chickens using them for different lengths of time each day. Some people regularly clean out their runs. I never do. Different things work for different people. I built my 12’ x 32’ run on a slight rise then built a swale on the upper side so the rainwater is diverted away from it. Part if it is covered with roofing from a shed roof that blew off in a storm but part is open. I mostly don’t use anything in it. The run bottom is clay soil. I did dump a couple of bags of pea gravel in the area where the run gate and the coop door are to help when it stays rainy for a few days. I know many people use all kinds of things, but I tend to stay away from organic things like straw, hay, wood shavings, or wood chips. I use that type of things for mulch in the garden. One purpose of mulch is to hold the moisture in. I want the moisture to drain out or evaporate in my run. I know others use this type of stuff and are very happy with it, but their management techniques and runs and conditions are almost certainly different from mine.
One exception to that is that I sometimes throw a bale or wheat straw in there with the baling wire removed. The chickens will have that all scratched apart in a couple of days, will eat a lot of the remaining wheat seeds in it and pretty much thresh the rest out with their scratching. Then I use that wheat straw as mulch in the garden. As long as it doesn’t set in wet and rainy for a period of time, it drains and dries pretty well. But I only do one bale at a time so it’s not very thick in there.
I don’t know anything about your run location or conditions, but if your sandy run is staying wet and yucky, I suspect it is not positioned where it can drain very well. Your problem may not be what material to use in the run but how to keep water out to start with. Any material is likely to be unsatisfactory. I suggest you read Pat’s article.
Hopefully you can get something beneficial out of this. Good luck!
