Dehydrated skin and lips are connected to two root causes in your body. One, weak liver and second, gut fermentation.
First thing I would do with severe case of dry lips and skin is take one tablespoon of castor oil in hot tea, be prepared to go to the bathroom, you will have a diarrhea. Then eat more fiber in your diet, no meats and sugary stuff. Take probiotics like kimchi, sauerkraut, even pay lose are ok. Drink plenty water.
One week later do a liver flush, this will take the burden from your skin.
You are lacking Magnesium, take 200mg Calcium and 500mg Magnesium together daily. Also take Zinc and Biotin.
Topically you can use ghee on your lips, honey, castor oil, coconut oil. I find chap sticks useless, but some natural ones may help you protect from wind and cold air.