I'm a very wet blower. In the winter water often drips from the bottom of my low whistles while I'm playing.
Condensation is an on going issue with any whistle, but more especially those made with metal. Some designs work a bit better than others, IME. If you take some time and view enough performance videos on YouTube you will find that everyone deals with moisture build up. You just have to realize that it happens and develop plans to take care of it. Some players suck! Some players blow! (That is not referring to their playing.) Some get the shakes and so on. It's reality and you have to deal with it, hopefully before it gets ahead of you and effects the sound and performance of the whistle. As long as your breath is warmer than the air inside and around the whistle there will be condensation. Winter temperatures make it more obvious as you have noted.
In my opinion, the Burke design works better than many whistles regarding condensation buildup. And it is one of many reasons that I continue to buy and play Burke whistles.
Just a caution if you are going to be a shaker. Watch where you point that whistle when you shake it out if you ever play in public.