I don't like using gloves to wash dishes either, but they make a HUGE difference. Dish soap is designed to dissolve and wash away oils and grease, which means this is happening to the natural oils in the skin on your hands if you're not wearing gloves. IMO, chemically stripping away natural oils just to try to replace them with something else is not the best solution (this is what we do with shampoo and conditioner), and having tried EVERY moisturizing option under the sun to fix my peeling hands, I eventually had to suck it up, use gloves and voila! problem solved.
That being said, as much as I love coconut oil, I don't find that it makes improvements in my skin or hair. What does make a big difference is honey, which is a humectant (it helps skin to retain moisture). I occasionally give myself a honey facial and nothing else has ever made my skin so soft (you can also follow up the honey treatment with a little coconut oil if you want, but I don't find it necessary).
For hands, I would put them under hot water to open up the pores, dry them lightly, then cover my hands with honey, and put on disposable gloves so I don't make a honey mess everywhere. I leave it on for maybe 20 min or more, then wash it off. That's the only thing that has made improvements in my dry hands when they were at their worst.