One more thing I do, but probably has nothing to do with your rust
After cooking, I leave the burners running for 10-15 mins to burn off any leftover stuff.
Before I fire it up the next time I scrub it with a wire brush.
By doing this I don't burn my brush on a hot surface, and it's easy enough to remove the leftover goo from the last session.
Mostly I'd say you need to use your BBQ more often to burn off any inside moisture.
I'm off to use mine right now.
Hand-made burgers mmmmmm
