That's pretty much all you can do. Try to dry them out, clean them, roll them up and forget them for the winter.
Next Spring when you open it, it will be full of water and have black mold everywhere and look like it's 300 years old.
If you cover the awning with the RV cover, it will help keep water out. But somehow, even with a cover, mine always managed to have water in the Spring, no matter how hard I tried to keep it dry.
About the black mold and such. When I started cleaning my awning with Mr. Clean Magic erasers, the mold problem vanished. I could roll the awning up in the Winter and come Spring, even if left uncovered, the awning was/is clean as a whistle. There's something in the Mr. Clean Magic Eraser that inhibits mold from growing. It's by far the best cleaner I've found for my awning to prevent black mold from forming.
Mr. Clean Magic Eraser? Always just wipe it nice and wet. Do not scrub. The follow up with a dry cloth and dry it. There's something about the chemicals that stops the mold. If you rinse it with a hose, you'll flush it all away, and the mold will come back. I don't know how it works, I just know it does.
Wipe, dry, done...small areas at a time.