Okay, this might sound weird but I suffered from the same thing my entire life...that was until I had my bite fixed.
The bone that the front teeth sit under might not be genetically where it's supposed to be (malocclusion type II) and will shove the lips (both upper and lower) into a position that they weren't genetically meant to be. I know this sounds far fetched....but ever since my bite has been fixed, my lips are in a different position (and so is my tongue and lower jaw) and I no longer suffer from puffy, dried up, peeling, followed by a day of raw lips. If your bite isnt in its correct place, you move your lower jaw constantly, and so do the lips. You end up constantly, all day long moving your lips around without noticing, even wetting your lips which makes matters worse...even if you're not a mouth breather, but it's much worse for them.
I have upped my fat intake through the primal diet and that's helped a LOT...but for the most part, the position of the bite and lips stopped this insane cycle.