D.B.
If it's constant, it's probably not "just dry skin" so I'd guess it's an allergy. You could try washing clothes in a dye-free, fragrance-free detergent, but if the itch is in places that the clothes don't touch, that's probably not it. Soap is drying anyway so try to limit bath time and don't use soap at all if he's not that dirty.
I'm going to guess that it's eczema which is usually due to a nutritional imbalance. Don't feel badly that you're doing something wrong - it is SO hard to eat "right" when so much food is processed and even produce/health foods are grown in depleted soils with fertilizers and contaminants. I have a bunch of friends whose kids got rid of allergies by using a nutritional supplement called Kids Now. You can buy it thru distributors who give you great help on an ongoing basis. It's patented (so no one else makes it, and what it says on the label is actually in the product!), safe for infants and pregnant moms, and made here in the US. One friend's 9 year old also takes a pinch of the fiber product from the same company and that seems to control the eczema. The mom had really bad eczema on her hands (they would bleed so much she couldn't even tie her kids' shoes). That has cleared up completely for her by using the adult product. She'd be happy to talk with you, and so would another friend whose kid had 50 allergies and they are all gone now. I'd be happy to give you more info and connect you to them.
I don't blame you for not wanting to give Benadryl a lot, and I agree that shots are a terrible idea in a little one. Besides, those things and even the topical creams are just treating the symptoms and not addressing the cause. You want to get to the bottom of this. You can spend a lot on doctors and tests, or you can try a supplement for a few months and see what happens. You've got nothing to lose!