One of the downfalls with eczema is the fact that it is different with each person. What works for me doesn't necessarily work for another person.
I developed eczema on my right hand that covered my palm and between my fingers and around to the top of my hand when I was 20. In the beginning it would flair up and then seem to almost go away....but it always returned. I called it my stress hand for awhile because stress made it burn and itch worse than anything I have ever experienced.
Jump forward 20 years, it got worse and worse to the point that it never went away...it was bad and then worse and then less than worse, but horrid just the same. At the worst cracks developed that were 1/4-inch deep and the skin would flake and bleed and it was painful to just use my hand for anything. As a cake designer/baker I ended up wearing gloves all the time, and that just made my hands sweat and made it all worse.
In those 20 years I dropped every chemical cleaning agent and went all natural for everything. I sourced every cream, lotion, over the counter and prescription. Steroid cream only took the edge off the itchiness, but never ended it...
My first step in the right direction came from using the homemade soap my aunt made. Instead of lubbing up with lotion after every time I washed my hands, they felt softer. The eczema was still there, but it was easier to live with. I used her soap for a couple of years before finally making my own. My breakthough didn't come until I finally started learning about herbalism.
Many simple healing herbs were weeds growing in my yard. Only after I started picking these herbs and schmearing them onto my hand and did it several times a day for weeks did I begin to see improvement. It took the entire summer, and I also infused these herbs into oil and made a balm to get through the winter. It took about 7-9 months total for my hand to look completely normal, no more red itchy splitting skin that burned constantly.
I now make infused oils for my soaps, and for hand salves. In my experience a good soap helped my skin condition but it didn't heal it. Once I was actually healed, and my skin had elasticity again, then soap alone helped keep it from returning. I have never had it flair up again. In the deep winter I might see a small patch of dry on the base of my palm and I immediately break out the infused oils for a day or so and it is gone again. I use a homemade salve and an occasional lotion, depending on how much work I'm doing and how I feel, but my days of never being an arms reach from the lotion bottle are gone.
Getting checked out by a doctor is always a good thing. The first thing they check for is liver issues, many skin conditions are signs of liver malfunction. Also they usually only prescribe steroid creams, which can't be used long term....and never helped me.
The very first thing I would recommend and the simplest is to infuse chamomile tea bags in almond or avocado oil in a gentle simmering water bath for about 3 hours. Cool, remove the tea bags, and store in the refrigerator. Remove a small amount each day and warm to room temp and schmear on the inflamed skin. Chamomile is a gentle healing herb and the oil is absorbed in the skin. If you have access to calendula, burdock leaf (spring harvest is best), plantain leaf (narrow leaf or regular), yarrow leaf or flowers and comfrey by all means use any of them or all of them.