I wasn't sure what kind of heat you had, but mentioned the wood stove drying the air just as an example that it may very well be at least part of the reason for Shadow's thirst. The others' suggestion about a senior panel at the vet is also worth doing. Assuming everything's normal, it might put your mind at ease and give you a baseline to refer back to in the future.
I don't know anything about space heaters or heater/humidifier combos and will defer to Dee on that.
I have central heat/air and have a humidifier in my kitchen and a small one that I keep by the head of my bed. I haven't been using them this season since it's been fairly mild (knock on wood), but I'm sure I'll be turning them on at some point.
Another tip, if you don't do this already: when the weather is consistently cold, you can set your thermostat for a lower temperature while you're sleeping, or when no one's at home, and warmer when people are home. The heat won't come on as often, so the air won't be quite as dry, and bonus, it'll help save some money.
I just realized: did you notice if Mango was drinking more, too, when Shadow was? He's younger so maybe dry air wouldn't affect him as much, but if his thirst drive increased, then that would really suggest it's the dry air, IMO.