Yep. DE is multipurpose, indoors and out- drying agent, bug control, dust baths...and yes you are correct about keeping the birds dry. If possible, your run will be nearly odor free.
Food-grade diatomaceous earth (DE) is useful as a drying agent and deodorizer indoors and out. So is Stable Boy. Combined with sand you'll see immediate improvement in your run. Ideally a covered run would save you a lot of grief, but it's an expense to create. Is your run covered in some...