Also look into handsoaps that have coconut oil and of course nothing called antibacterial. Never let dish detergent touch your skin. Use nonlatex gloves to wash dishes.
First thought is are you wearing latex gloves? I have psoriasis since getting my gallbladder out, which demonstrates the connection between digestion and skin. I would try to improve my diet and take things like magnesium , vitamin B6, and look into nutritional connections.