Wow, Clint, you are reading my mind!
What exactly are your symptoms? In other words, what problems are being caused?
For my part, I'll notice a bubbling sensation. It's weird, but it doesn't seem to affect the tone in any way. Because of that I suspect the problem is happening at the bottom tube plug - water gathers and fills the little hole of the plug.
What I've done in this case is to always have two plugs and two mouthpieces and swap them out when one gets too wet. But I'm panicked that I'll drop the tube plug and it will get lost on a dark stage!
I mean, this kind of thing has always been a problem with wind players (as I'm sure you know), Usually, people don't mind taking a break while someone swabs out their instrument.
Would also be worried about blowing the thing down the tube, but "nary a drop" sounds really good :)
Search "hydrophilic" on Amazon and you get something else that might work - here