The white pus seems to be a fungal infection. But your is severe, and due to the shaving the hairs have turned back into the skin and cause the redness and inflammation. You can use head and shoulders with pyrithione zinc or Nizoral shampoo. Then take some cortisone cream and some anti fungal cream like lotrimin AF is best and apply twice a day. Ohh the shampoo is used kinda like a mask for 10 min if it irritates 5 minutes and u rinse it off. That was happening to me and I struggled for months and after 4 failed dermatology appointments I discovered the answer on my own..
Any time you take an antibiotic for something that won't actually kill you, there are big risks. One is making the problem chronically worse, as they will only work while you use them and then you go back to being who you were before but possibly worse as antibiotics kill off beneficial organisms that protect us from such things. While acne is annoying, it won't kill you, so antibiotics would only be appropriate if the acne was getting in the way of your life. You also have to consider that the products doctors recommend for cleansing often have a lot of synthetic additives in them and are often also antibiotics in part, so again, temporary relief with the possibility of more long-term harm. As for nose being oily, oily skin usually is where we get acne, so that's not it. It might in fact be that the nose sticks out and gets a lot of sun, and ultra-violet light dries out the acne. When I had acne when young it went away pretty much in the summer because I got a ton of sun, which is now not recommended because of skin cancer, but it was a lifestyle where and when I grew up. If you really want to experiment, try radically changing your diet, to start with. Avoid dairy, wheat, sugar, chocolate, and the like and see if the problem gets a lot better. If it does, you have a food problem. And try using more gentle natural cleansers that keep you clean but don't add to the problem. You want to keep those pores from getting clogged, so stay away from anything with petroleum derivatives in it, and watch the oily foods you eat -- just the good fats, not the bad ones, avoid fried food, etc. If that works, it will keep working, not stop as soon as you stop the medication. If it doesn't work, nothing lost. Some people just are unfortunate in that they have very sensitive skin that needs a lot more care than others. My brother was one of those, and he grew a beard to avoid the problem, at least on his face. Get out in the sun when you can. Peace.