Alchohol most definitely stings and is harmful to tissues. I've stuck my fingers with fresh scratches and pokes into jars of alcohol cotton balls way too many times at work for you to tell me that it doesn't sting. It is a poor choice for wounds or skin infections.
Sillybunny, cortisone and antibiotic ointment has it's place in treating wounds but it shouldn't just be applied to anything. This combination is used to prevent proud flesh in healing wounds but it is going to slow healing somewhat and increases the risk of infection because the cortisone stops the body from fighting off infectious organisms like bacteria or fungi. So, it's not a good idea to just put this combination on any wound. Plus, if this is a fungus then an antibiotic cream isn't going to help things and the cortisone will keep the body from fighting the infection.
Betadine is both antifungal and antibacterial, that is why it is commonly recommended as the first course of treatment for skin issues and is used for cleaning wounds. You can get betadine solution to use during the winter so that you don't have to rinse it off. Just apply it to the affected areas (not right next to eyes) daily and see if the condition begins to improve in a couple of days. If not, contact your vet.