Can you get an awning?

Check out PlastPro composite doors. They are made from non-wood materials and designed to resist water damage. http://www.plastproinc.com/

Have you looked into fiberglass doors?

An awning isn't possible unless I remove all or part of an overhead trellis(?) that is adjacent to and nearly touching the shed. But I may have to consider it. One way or another, the doors will need to be replaced. I have a front gate that's fiberglass, only 1.5 years old. It's gorgeous, but I do notice that when there is heavy rain, it expands. Will look at the Plastpro now. Thanks!

I just looked at the Plastpro site and I will definitely go by the dealership to discuss it further. I think I need to get an awning in there, too for extra protection.

@critrnurse I also highly recommend you install a metal head flashing. A licensed contractor or experienced carpenter will know what this is. It protects the top of the door from water running down the outside face but also any that gets behind your finish (siding, stucco, etc.) Here's a great article to read: http://www.finehomebuilding.com/2016/11/29/adding-head-flashing

You've got a hack contractor if you keep having the same problem.