About a month after my move from Arizona to New York, my hand have dried out exponentially and around my knuckles have started to become a dark red. They feel gritty when I rub lotion on them and dry out almost an hour later.
Here is a picture. It's on both hands.
It might help to use Vaseline. Sometimes clients skin get's irritated and lotion doesn't help; I use a Vaseline type substance. You can also try Sensi-Care if the Vaseline doesn't work. it is basically half Vaseline with some some other things (15% Zinc Oxide used for diaper rash) I however am not 100% that your hand is doing this because of the cold. Good luck!