This sounds a lot like eczema
. Eczema is an inflammation of the skin which often has a complex set of causes, including your personal disposition to have sensitive skin and exposure to various environmental irritants. It can be difficult to treat because of how frequently the hands come into contact with irritating things.
However, some basic strategies can help. First, minimize washing your hands and never use very hot water; this will keep your skin's basic layer of protective oils intact. Never rub your hands dry, pat them instead. Apply a thick moisturizer multiple times a day to minimize itchy and dryness. If possible, where gloves whenever you are washing dishes or come in contact with cleaning products. Avoid any strongly scented or otherwise harsh soaps or personal care products. You may also want to try applying an over the counter anti itch or steroid cream to the most severely affected areas, and you could try wearing gloves at night to keep yourself from scratching.
If none of this works, then you should go back to see your dermatologist
again. There are numerous stronger steroid and anti itch creams that can be used to cut down on the itching and inflammation and help your skin heal.