For seasoned players the problem - ie the moisture - will be confined to the mouthpiece. In this case, better than blowing through the whistle with the window covered is to close all six holes, and also the tube end, then suck sharply through a) the playing end and b) the window.
If you think better of your lungs than to use them as a filter like this, blow instead. but only through the window - blowing through the mouthpiece will make a hideous din. Sucking is best, as the air drawn through is drier, but blowing is still quite effective - you will shift much more than you deposit. But before blowing , spare a thought for where the mouthpiece is pointing.
Depending how large your hand is, you might be able to close the tube-end with a little finger. Those with smaller hands, or trying the technique on a Low D, will need to use their knees instead. (In the case of a Low D, some contortion is called for as well.)
Slightly complicated, but highly effective as a short-term fix. Done regularly it will certainly stave off the day you need to attack your whistle with a pipecleaner. (Perish the thought!)
Or you could simply use a Delrin-lined whistle. My Burke D is thus lined, and it makes a huge difference - I go 30-40 minutes with no clogging whatsoever. I think some other makes are similarly lined these days - maybe the Sindts?