The problem with sand is that the poop filters down into it and never goes away -- stinking every time it gets wet.
People who have success with sand almost always live in a dry climate, use sand in a roofed enclosure, and scoop it daily.
What's the source of the water that's causing the problem?
If it's rainfall then putting a roof over all or part of the run can help.
It it's rain from the coop roof or the roof of another, nearby building you can divert it with gutters.
If it's groundwater flowing through the run area you need to dig diversion ditches, make grass swales, add French drain, or otherwise re-grade the ground to send water around instead of through.
If it's a low spot that collects water there's not much can be done.
Many people find that a good layer of wood chips -- the kind you get from tree trimming -- are the best option for keeping the chicken run from getting muddy.
Given your location, I know that you have pine straw available (I used to live in Millsboro, DE), either for the raking in your yard or commonly available from landscaping supply places. I LOVE pine straw for my run!
It's not as absorbent as some of the other options, but it never packs down and the surface dries out after rain as soon as the sun comes out.