+1 on the shea butter.

It might be helpful to learn a bit about surfactants so you can pick a gentler hand soap. A lot of what is being called "soap" on this thread are really detergent cleansers, such as sodium laurel/eth sulfate. This stuff will strip the heck out of your skin if you use too much (very common with liquid hand soaps) or it's not mixed in with a good conditioner/moisturizer. SLS and a few of its related cousins are the surfactants found in most shampoos, hand and bar soaps, dishwashing liquid, laundry detergent, etc. It's what we all started using when we gave up lye soap. Incidentally, we didn't make that mass change because detergent cleansers are any easier on our skin, because they aren't. What detergents do better is rinse away in hard water conditions, which are very common in the US; they're easier on our clothing, plumbing and appliances.

"Real" soap, which is saponified with sodium or potassium hydroxide, had an underserved reputation for being caustic. A hundred+ years ago, this was true for most homemade soap because it was difficult to control the strength of their homemade lye. Today almost all handmade, high-end soap is made with commercially produced powdered sodium hydroxide (lye) or potassium hydroxide, which allows us to tightly control the amount of fat/conditioners in the finished soap. You really might consider trying a superfatted handmade soap, which you can pick up at just about any natural food store. It's more expensive, but if you just use it on your hands and not in the shower it'll last forever.