Hi. seems you have been developing "Cummulative irritant contact dermatitis" over your hands as a result of repeated hand washing with the handsanitizer. Its due to excessive drying as a result of it. Handsanitizers are mostly alcohol based and totally dry out the skin and are responsible for this very common occupational dermatosis
which is also common in food handlers and in people who do household work.
Apply a good moisturizer every time you wash your hands. For the dry patches over your knuckes you can apply a potent steroid
like tenovate cream twice daily over moisturising cream.