Even with the moisturizer, hand sanitizer will cause my knuckles to become so chapped they cracked and bled within a few days with just a few applications per day. I had to stop using it as I began to fear infection.
Yes, the moisturiser mitigates the issue but will not solve it. If you need to sanitise your hands a lot and they’re sensitive you may want to use or switch to disposable gloves.
Or a well chosen brand of non-disposable gloves that survive disinfection with it well. Nitrile and butyl don't and are good for incidental contact only, they will swell; latex, vinyl, thick PET and norprene do work for more extended time. You have to be thorough in rubbing so mechanical resistance to that could be important too, making norprene the material of choice closely followed by PET.
(The latter three depend on the brand, some might start to slowly dissolve or harden after many applications due to plasticizer leaching. Always check with manufacturer.)
In the 80's they started making alcohol based sun screen. I thought the stuff was great! Yeah not great. I'll still use it but only for one day. Otherwise I use the greasy stuff.
This is one reason commercial hand sanitizers are [usually] made predominantly of ethyl alcohol, with a small amount of water and IPA added in. Ethyl alcohol does not dry out your skin nearly as much.