During work I use an alcohol based hand cleaner with Aloe. The dispensers are positioned at the entrance inside each room and in my office. After seeing the patient, I hit the button as I leave the room. Yes, my hands get dry and chapped with repeated hand washing. When I retire for the day, I hit the Vaseline Intensive Care hand lotion just before I go to sleep. Keep it on all night long. In winter, my skin dries out extensively due to hot showers. After I get out, shave, etc., I use the moisturizer on my lower body and the extra goes to my hands and arms.
For really bad chapped hands, I fill a pair of exam gloves with several squirts of Vasoline Intensive Care lotion and wear them while I sleep. My hands are slimy when I wake up and remove the gloves, but the excess is easily washed and the irritated, chapped skin is resolved.