Dr. Bronners and other liquid castile soaps are too drying for my family's skin. We prefer Dr. Bronner's bar soaps.
We keep a bottle of good moisurizer by each sink, for use after washing. My personal habit is to wipe any excess off my palms - it's the back of my hands that dries out.
Burt's Bees makes good products, but cheaper alternatives work just as well. Suave Oatmeal, Suave Advanced Care and Suave Sensitive Skin are gentle and effective. None are heavy. Suave Oatmeal is the simplest forumation and is without fragrance or dyes.
You mentioned using Dove. If you like Dove products, try the liquid hand soap for dry skin (not the body soap). The hand soap comes in a white pump bottle and is much richer than the body wash.
FYI - Dove, Suave, and Vaseline Intensive Care brands are made by the same company. From a position of knowledge, I tell you that the difference among brands is less about formulation and more about packaging, marketing and fragrance (in scented items). You won't be compromising on effectiveness by buying Suave moisurizers. Unfortunately, Suave doesn't have an equivalent for the the Dove liquid hand soap.