I have eczema and my hands can be terribly dry this time of year. The most recommended moisturizer - Vaseline. It feels a bit odd at first, but it's the only thing I've found that keeps my hands supple while not causing irritation. Also, wear dish gloves for washing up. In fact, with eczema, I wear cotton gloves inside my dish gloves because hands sweat inside them, further irritating already sensitive skin. I've also started buying bars of glycerin soap (available at Bulk Barn) and it seems to be helping too.

I know all too well about having sore hands. Hope this helps!