I don’t know if this fit’s the description of what you’re talking about…but dry/cracked/sore skin will sometimes appear on the corners of your lips. It’s called Cheilitis. In general, Cheilitis, is inflammation of the lips. I once asked a doctor, who advised application of vaseline…and taking B vitamins also helps (I’ve found). Cracked skin on slips can be triggered by a vitamin B deficiency sometimes.
So, keep applying Vaseline. Or try olive oil even. Keep well-hydrated. Drink lots of water…because lack of water makes lips dry. And consider taking B-Complex vitamins…or maybe multivitamin. Or to be on the safe side…consult a doctor.