Hand sanitiser can cause all sorts of skin issues, unfortunately...
This is mostly due to its very high alcohol content: sanitiser needs to be at least 60% alcohol to be effective against germs, but alcohol breaks down the skin’s protective barrier of natural oils, and allow allergens into the bloodstream.
Once this happens, it can trigger an immune system response, causing inflammation, itching, itchiness and rashes.
So while alcohol-based hand sanitiser is often responsible for sore, dry skin, for some people it can be a more serious problem. That's because there are ingredients in the sanitiser as well as alcohol, and those ingredients (fragrances, dyes) can act as allergens. For those people, hand sanitiser can cause serious irritant contact dermatitis, or allergic contact dermatitis.
If your hands are swelling up, cracking, peeling, weeping or exhibiting other signs of an allergic reaction, see a doctor or pharmacist, and stop using that particular sanitiser. It may be that you can find one that doesn’t contain perfume and dye, and which your skin will tolerate better.
In the meantime, switch to washing hands thoroughly with soap (or a soap-free wash) and warm water, rather than relying on hand sanitiser. This is a better practice than using hand sanitiser, except when you’re somewhere without running water. Soap physically binds itself to the germs and is washed away as you wash it off.
And make sure you moisturise thoroughly throughout the day!
Don’t moisturise immediately before or after applying hand gel, as you want the sanitiser to get to the germs and kill them, but do be generous with the balm as often as you can.