Yes, water will seep in the awning even when rolled up.
Clean the awning with a good cleaner/mold remover.
The best stuff I have found is JOMAX Cleaner & Mold/Mildew Remover
Mix it up and apply with a garden sprayer. It really works!
(great for around the house too)
Then once clean, periododic treatments with something like 303 Protectant will keep it looking brand new for a long time.