Jani, my location is described exactly as yours and although I don't get bloody noses at altitude/dry, our son does. He uses Bag Balm, which you can buy at CVS in a "cubular" green tin (or at tack n feed if you head east to Poway, Santee or Ramona). We all use it on chapped lips.
He uses a q-tip to line his nose with bag balm when we get to town/trailhead, such as Mammoth, and then again the next morning before we head out. This has been enough for him in moderate weather, but you could bring a glob and a q-tip along on trail, obviously.
If you don't know, Bag Balm is technically for udders, ie cow bags, and is a petroleum jelly/lanolin formula with another ingredient using polysyllabic format (that I'll preserve for someone smart to copy/post)but is used for chapped hands, lips, skin, nether regions, etc... by a lot of people. Good stuff and that green tin will last you a long time.