Sorry about the dry skin. You're right, hand sanitizer makes it even worse.
I used and continue to use Aveno Skin Relief Moisturizing Lotion - Coconut on my hands throughout the day and on my feet at bed time. If my feet are really dry I'll put the lotion on heavily and then pull on a pair of socks.
Keep an eye on your scalp as well as it tends to get dry. Little bit of coconut oil at bed time before putting on your night cap helps with that.
I had my second round of chemo on Monday. I use Lubiderm unscented moisturize. I apply it 2 to 3 times a day especially after I’ve showered. I hope it’ll help, if you try it!