DE can be a desiccant, it will dry things out. But once the DE gets wet it doesn't work any more. I don't think DE is the answer, it is no longer effective when it gets wet until it dries out again.
You are experiencing something pretty common when an organic material like straw, hay, grass clippings, wood chips, wood shavings, dried leaves or chicken feed stay wet a bit. Mold grows in those conditions. It doesn't matter if it is in a compost pile, a pile of dead leaves, a chicken coop, or a run, if it stays wet it can mold. When it sets in wet it's really hard to keep a run dry, even if it is covered. Rain blows in from the side. The ground itself can mold if it has something organic like chicken poop in it.
Some photos could help a lot in knowing what you are dealing with, both the two kinds of mold and what they are growing in or on. It's easier to respond if we know what you are dealing with.
There are a huge number of different kinds of molds. Most of them are not harmful. My chickens love to scratch in leaf mold, that's where dead leaves pile up, often in fence rows, and can get moldy. It doesn't hurt them. With some harmful molds, the kind that are a danger in wet feed it's not the mold itself that is harmful but the secretions as it grows and multiplies. Especially if those are in a container they can quickly build up to a dangerous level. That's why moldy chicken feed can be dangerous. Some molds can be dangerous, I'm not suggestion that you just blow them off. I think they need to be dealt with. Just don't panic when you see a mold. Molds are all through nature.
In general the best way to deal with a mold is to dry it out. Not always easy. The spores will always be there, when it gets wet it will start growing again. What's the best way to dry it out; depends on what you are working with. Raking might be enough, removing wet bedding might be best.
If your run is a low spot where water collects you are going to have a lot more problems than if it drains well when it gets wet. If your run is collecting water we may be able to help you dry it out. A lot of this depends on what you are working with.