While it's prudent to wash your hands out in public to avoid picking up germs if you tend to put your hands in your mouth, or handle food that goes directly into your mouth after touching things strangers touch. You stand lest risk in your own home. Still wash after going to the bathroom, but otherwise at home, unless you're handling questionable foods like raw chicken, you shouldn't worry as much. Also, the amount of time you spend, the heat of the water, these can stirp your hands of essntial, protective oils.
I simply rub petroleum jelly on the backs of my hands especially at night when i won't be washing it off immediately, you can slather your hands with it and put on thin clean cotton gloves at night if your hands are in really bad shape, you will be astonished at the improvement in the morning. But if the cuts are open and bleeding like dl's I'd dab neosporin on them.