Your mom was right. She knew her stuff when she told you to wear gloves when you wash dishes. I get it. Gloves can be annoying and kinda make us look like a 50s housewife, but trust me when I say it's totally worth it.
I bring this up because recently some of my clients have been asking me for more skin creams for their overly dry hands claiming nothing is helping. I LOVE Hale & Hush's Saffron Meristem Cream for dry skin and don't get me wrong, I'd be completely happy to sell multiple bottles of the magic stuff to everybody. But along with amazing skin care products, one of the easiest things you can do to protect those hands is wear gloves while washing dishes, cleaning and even cooking.
If you wash dishes with ungloved hands, you are also washing off the natural oils, (aka protection). Harsh soaps and hot water can cause major issues in time with dry skin, rashes, dermatitis, and irritation. And keep in mind that a lot of us have been using tons of hand sanitizer due to the plague and all, which is harsh and very drying for the skin.
What I like to do is wear Nitrile gloves anytime clean, cook, or wash dishes. If I am doing anything where I am going to be washing my hands multiple times or exposing myself to soaps, etc, I use gloves. Then instead of washing your hands, you are washing gloves. They can take it. Nitrile gloves are the medical type gloves you have seen. Nitrile gloves fit snugly so they really don't get in the way. So, glove up and watch your dry, itchy, rashy hands become healthy and hydrated.