More important than switching to sensitive skin detergents, would be to switch to products that are plant rather than petroleum based. Sun & Earth, Ecover and 7th Generation are good choices. I make my own lotions so I know exactly what's in them, but you could use coconut oil, olive oil or vitamin E oil straight out of the container as a moisturizer.
Your condition is an allergic reaction and you will need to find the cause. Traditional medicine will alleviate symptoms but not necessarily cure you until you get to the root of the problem. I would start with elimination diets.
I went through severe eczema about a year and a half ago. While what happened to me may not be the answer for you, maybe it will shed some light. My body suddenly started reacting to dairy. All allergy testing came back negative. What was actually going on (because hindsight is 20/20) was sludge in the gallbladder making it difficult for my body to process many fats with dairy fat being one of the biggest culprits. It took about 8 months of elimination diets and herbal liver cleanse teas to get through it. Liver herbs are burdock root, dandelion root, yellow dock root and milk thistle seed.
The one thing that cleared things up the quickest for me...mega doses of multi-B vitamins. B vitamins are for stress (and your body is definitely stressed right now). It is water soluble, so you cannot take too much. Your body will eliminate what it does not need through your urine.
Get a full blood panel done and take a close look at your liver enzyme numbers. I'm betting they are probably high right now.
Best of luck in your healing journey!