Every winter my hands crack with dry air and frequent hand washing. This is how I address it.
Amlactin lotion will moisturize and eliminate the dry skin. Lotion up. Put latex or nitrile gloves on over the lotion. Wait thirty minutes or more before removing the gloves and washing your hands. Doing this every week or two will moisturize them, keep them soft, and prevent cracking.
I wash my hands at least a dozen times a day and use hand sanitizer at least another forty times a day. The gloves and Amlactin lotion help me survive each winter and have protected my hands throughout the pandemic.