Oh man do we deal with sore, chapped skin around here! I have one daughter with sensitive skin and another with eczema! What I suggest for chapped skin is one of two creams - Barrier Cream, found in the baby department with the diaper cream, it isn't a diaper cream but it does work to help soothe the skin. The other is Glaxal Base cream, it's completely fragrance free and soothing.
Then the best thing is to apply it at bedtime, this way they are asleep and it can do the job without being rubbed off immediately. We also used cut up receiving blankets to wipe their noses. You can wash them right away ( I wouldn't use the same one all day) but it's way gentler than a tissue. We always say "Hold still please, Mommy with be gentle" and wipe really slowly. Holding their tiny heads still to do it quick ends up being rough. Sometimes that unavoidable but slow and careful helps keep the ouchie away.
Good luck! It'll be better soon!