Cetaphil instead of soap and lots of moisturizer. Find a moisturizer that doesn’t contain alcohol or scents and try to use it after each hand washing. Like @Nimis said, putting the moisturizer on under some gloves (avoid latex which may irritate your skin further) works great. Do it before you go to bed and try to keep the gloves on all night.
My kids get horrible dry skin in winter, and the pediatrician said the harder it is to work a moisturizer into the skin, the better it is for you. Eucerin is great. Also try Aquaphor, which is more like an ointment (made by Eucerin) and works really well on chapped, cracking skin.