Alcohol will dry your skin and that is most likely what it is coming from. If you want to continue to use alcohol, then you need to get a good moisturizer. Ask your pharmacist if there is something else you can use to sanitize that wouldn't dry your skin out so bad. The dryness will make it itch.
. Alcohol removes the oily protective layer of the skin, leaving it exposed to surfaces and the elements making it more prone to breaking down. The chemicals will also be damaging to skin cells causing them to break down and die and may result to an inflammatory
There are some alcohol available with moisturizers in it. You may want to take a look at these. Another option is to moisturize after handling your baby or at night. Take care and regards.