from your above description, use of alcohol over the dry patch has caused an contact irritant reaction on your skin.
apply a calamine
lotion mixed with emolient , as it is a soothing agent and will be helpful in reducing itching.add an oral antihistaminic iif it itches a lot for a few days.
Use a mild steroid
like hydrocortisone cream
over the involved area . but not for longer time as it can cause acne
advise you to meet a dermatologist
for clinical examination and specific treatment accordingly