My daughter has had a severe case of this her whole life, all the same spots you mention. It is now improving at 3 and I've heard from many people that it will pass someday.
We do have one cream that really works:
Triamcinolone Acetonide Cream USP, 0.1%
Like you, I avoid medications, so since the cream is medicated and prescribed by our doctor, I only use it when it gets really bad. But it instantly improves. If I put it on at night on a super flare up, it's way better in the morning, and if I put it on one more time it's almost gone. Then I don't use it again until it's bad, because it's chronic and can't be cured, but technically if we used it all the time, it would keep it totally in check. The label says you can use 2-3 applications daily, and if we did that, she wouldn't even have the ecxema, I just hate to rub drugs into her skin.
I don't even like to use regular lotion on the kids, I like to let their skin be itself, and no other lotion we tried worked anyway. Natural things like olive oil and coconut oil moisturized but did nothing for the issue.