I’ve just started with the problem . . . but I’ve been drinking a lot more water for health reasons. Before that it was only tea so maybe tea dries the mouth out (but I only drink decaffeinated) and I need to go back to having a cup of tea before playing. Seems a bit ridiculous. Is there such a thing as blackthorn tea?
-Does playing “The blackthorn stick” have the same effect? 
I’m glad you’re not having this problem with flutes.
I’m not taking the medications to make my mouth dry, that’s a side effect.
If your tongue touches your back teeth, your mouth thinks you’re eating and will produce saliva. So - what if you try relaxing your tongue (and anything else that feels tense). If inside your mouth you make the shape eeeee-eeee when you play, it kind of makes your tongue touch your teeth. Try out different vowel shapes, see if that helps! - Lesl
I can’t think of anything except 1) don’t trigger saliva production and 2) don’t let the tip of the whistle mouthpiece contact the inside of the mouth, so that the condensation in the windway has to come from moisture in the breath, not saliva, because saliva would have no way to get there.
Alternately,
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join the USA space program and only play whistle on the Moon’s surface with your helmet off, it’s very dry there,
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find some thick winter socks, cut one up and stitch together a round donut-shaped moisture-absorbing ring to put completely around your lips where they touch the whistle, for a 4-season moisture proof comfy fireside whistle session. Give it a name, mass produce it and cure the world of this major problem forever.
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build a helmet with a battery and a computer fan aimed at your face to dry out your mouth. The techies in the audience will automatically want to buy all your albums.
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ever considered the many benefits of playing drums? You can have a spit bucket right off the side and it never impedes your playing.