jgreiner - None of it is spit, man. Well, maybe a tiny bit. It's condensation from the moisture in my breath. I'm not blowing gobs of saliva down my horn for crying out loud! And it's not even all that much, no more than I see in any other wind instrument, but the bubble thing happens nevertheless.
FredCDobbs - No offence was intended on my end either. My question, however, was not "Oh no what do I do my fingers are wet?" but rather "Does anyone know of a way to modify a horn so that the water stays inside it rather than coming out the toneholes?" Which also doesn't seem like quite as a ridiculous a question for an experienced player to ask. Perhaps people are responding to the title of the thread and not actually the question I asked. Which is understandable, we all skim from time to time.
Gordon - no, only rarely using a stand. Why do you ask?
Also thanks Gordon and DrG for the suggestions. I'll look into that.