If you are putting rubbing alcohol on the piercing, that may be adding to the irritation and problems. Alcohol is very harsh and drying, something that retards cartilage piercings (i.e. nose and upper ears) from healing.
Quite frankly, most body piercings do not respond well to multiple, aggressive treatments. This tends to only add to the inflamation. Try the most gentle care you can for it. Make sure that when you wash your face or bathe, that all soap is well rinsed out of the piercing. Unless it is actually infected, using antibiotics can irritate it.
An infection is characterized by redness and tenderness that keep increasing, by weeping and discharge that is yellowish or pus-like. It sounds like you bumped it, and triggered a friction reaction, whereby the piercing is trying to form scar tissue to protect itself. This scar tissue is called a keloid. It it bleeds when you clean it, the piercing is NOT infected. Infections do not bleed. It is irritated and unhealed.
Noses take 3-4 months on average to heal and if you have a snag or bump, it can break open the new tissue inside the nose, sending you back to square one. If you keep snagging a ring, wearing a stud may help, but studs also tend to make the piercing harder to clean and prevent them from getting fresh air.
Instead of talking to your doctor, you might want to check in with your piercer. They have more experience with body piercings than most doctors.
