Cream/Butter type products might work better than lotions. Try etsy or Health Food stores for nut free body butters.
You can also try using a cream that's meant for heavy face moisturize on your hands, especially if you have some leftover from a previous purchase that didn't quite work out for your face. I don't mind my hands/legs being a little shiny but my face is a different story.
You might also want to switch the soap. Most soaps use sodium laureth sulfates to foam up and those make my skin extra dry. Oil based soap might be better (nut free ones, of course).
Olive oil does work pretty well. Most shower scrubs contain a grit like sugar and vegetable oil like olive or grapeseed oil. Avocado oil is extremely moisturizing but a bit heavy. Jojoba is also popular. You can try olive oil and a pair of old socks on your hands.