Hopefully you checked out Pat's page on fixing a muddy run. It has good info.
I don't know how big your run is, how well it is covered, or the lay of your land. Some, or maybe all, of my comments probably won't apply to your specific run. A lot depends on just how wet if can get.
Putting a few more inches of a sandy soil will certainly help with drainage. If you build the level up with a clayey soil, it will help keep outside water from running into the run and getting it wet, but the chickens will scratch holes in it that can become mud puddles if water gets into them. Sand will not stay wet like clay will. Sand or a sandy soil is certainly better.
Sand may need to be contained. That means a strip of something to help keep it from washing out and being scratched out of the run. Probably more important in a smaller run that a larger run and certainly depends on your layout. My run is on a crown. The natural slope is on one side and I created a swale on the other to keep water out. Even if my chickens scratch out my clayey soil, water will not run into it. Now, mud puddles from blowing rain is another problem.
With a covered run, a little strategic ditching and maybe gutters and downspouts may keep a lot of outside water out, but you'll get all that from Pat's site.
The reason I don't like organic material like straw, wood chips, or shavings in a run is that if they get wet, they can get moldy, which is not good for the chickens. As it rots, it turns to a black organic material that is great for your garden but can contribute to a messy run. One reason compost is so good for your garden is that it retains moisture. Straw especially can mat down and be hard to remove. If your run stays dry, the organic material gives your chickens something to scratch in and find more bugs. By scratching more, they will keep it stirred up, thus mixing in their poop better. You can certainly try it and see what happens, but I'd initially try a thinner layer of organic material that may be easier to deal with if you need to.
Good luck!