My nose gets really dry; the passages appear to be red and swollen. What can I do to remedy this? I've tried nose sprays; they make things worse.

Saline nose spray. Dry nasal passages can be due to dry ambient air in the home (get humidifier), meds (even non prescription, check with doc), dehydration (stay well hydrated). Can try nasal saline sprays (watch out for neti pots, some cases of fatal fungal infections-must be very clean). Avoid otc decongestant sprays (have rebound effect, habit forming, cause other problems). No relief: see md.
Ponaris oil. There is a wonderful otc product which is magic for dry angry noses. It is called ponaris oil. It was designed by nasa scientists for astronauts who all complained about dry angry noses when they flew. It is made of medicinal plant oils. If your pharmacy doesn't stock it you can have them order it or get it on line.