Today is a perfect example of why you should drink more water. Zero percent humidity.
Cold dry air will extract moisture from your skin, hence the dry lips and the cracked hands.
It may not leave you feeling thirsty, but you will lose more water this way and so you should compensate by drinking more than you normally would (and consider wearing gloves when outside to protect the skin on your hands).
I've spent a lot of time in cold climates, Living in snow holes, biathlon skiing, recreational skiing.....and I'm from Scotland.... Don't worry about looking cool, wear a balaclava if you feel the need. Remember, any fool can be uncomfortable.
If your nose is blocked/streaming and you are breathing through your mouth when outside, the air will be taking moisture from your mouth and throat too. Drink more (and shut your gob if you can) :)