It certainly doesn't hurt to add some moisture to the air. If you have a wood stove or pellet stove, you can put a large pot of water on it and that can add some moisture until you can find a humidifier. But for what it's worth, my parents' house had a fireplace and then a wood stove, which they used regularly to keep the heating costs down. And let me tell you, it was DRY in that part of the house, and yes, it made me very thirsty. That dry air is going through the nasal passages and down the throat (?) to the lungs, so it can dry all of that out. That may or may not be what's going on with Shadow (I really hope it's something as simple as that), but like I said earlier, it won't hurt either of you to add moisture to the air while the heat is on.
I hope this helps!