Letting them dry out prevents moisture from starting fungal problems in the future. But if a sudden drop in temperatures is in the cards for your region, I would prune the leaves now and see if they dry faster that way. You can always set up the cage early (early October say) and keep the leaves in plastic bags until you need them. We have had ooodles of rain here too. Only the dogs do not care when walking outside! Hee hee hee!