Those are some really beautiful leaves. Nice job. If you can't find carpet, you can lay down some plastic sheeting to make clean up easier and spread some hay or straw or leaves or anything that will hold water. Try to get a Hygrometer(?) so you can know exactly what the humidity around the leaves truly is. Going by the weather report won't work. I see you have your leaves in bunches and that's good. They share the moisture from each other. If you go back to the shed, bunch the bunches even closer together than is shown in the picture. Is your shed open or closed? Use your thermometer and hygrometer and experiment with closing the shed up. It will help hold moisture, but may also drive up the temps. You'll just have to play with that. If your humidity goes up at night, definately open up the shed and take advantage of it. It's the average humidity you have to think about. Misting the leaves can help, but I think like Johnlee said, your best bet is to get something that will hold alot of moisture onto your floor, and play with opening and closing the shed during higher humidity and lower temperature times.
Is the shed insulated? If not, you can lower temperature several degrees by using a lawn sprinkler on the roof. Set the pressure as low as you can go, trying to achieve a balance between good coverage and not wasting water.