I overslept this morning--opened the coop about 50 minutes later than usual and mine went straight for the water. Maybe birds are just thirsty in the morning and something I've not seen before because I open the coop before they are ready/light enough to go out.
They won't be drinking (or eating) at night. If really proofed, the run will be fine. It might be a ventilation issue causing the bee line run to the water and in that case the run would be preferred.