When I lived in Las Vegas and Reno, I used the plain old strawberry Chapstick brand all the time and never had any problems.
As soon as I moved to Columbus, OH, the regular chapstick stopped working for me and my lips got SO chapped that I'd have this red, sore, peeling area all over the skin just above my lip (almost like a red, sore mustache). It was terrible.
I finally started using Carmex (the kind that comes in the tube; I also started smearing some of it over the top of my lip where the red part was, too) and it stopped happening. The red part went away and I've been using Carmex ever since (although I had to get used to using the goopy tube stuff instead of the nice, neat stick that I was so used to before this).
I thought about trying to go back to my tried and true Chapstick brand, but I'm worried it'll start happening, again.
Good luck!