Agree. If putting in a thick layer of sand now is not an option then I'd be looking at ways to keep rain/water out, including getting the run covered and re-routing any runoff away from the pen. Actually I'd be doing both, regardless, especially with ducks.
And no, you are not overthinking...
I honestly wouldn't use either. If the run is very wet and muddy I think you are going to be very unhappy with either hay or straw in there very quickly. It's going to become a soggy, wet, stinky mess that will be a real back breaker to clean out. Wet hay/straw will also get moldy.