I think the moisture is not saliva but the moisture which is in your breath (the same moisture which in cold wheather makes you look like you are steaming). As the whistle is usually colder than your breath, the moisture is condensing on the surfaces of the whistle (same effect as you have when you breathe against a mirror).

If - as foolestroupe suggests - the whistle is pre-warmed up before playing, the moisture does not condense so easily, so the effect is not as strong.

I also noticed that if I play for a longer time, the effect gets less and less - probably because the whistle gets warmed up by the breath during playing.

More powerful lungs won't help - as you noticed yourself stronger blowing just leads to higher, not louder, notes (ok, there is a small range of loudness which you can control by blow strength but it is not very much).