Mine have been like this for what seems like forever. I don' t have a quick fix for you, but if the medication is making things worse, I would stop using it.
Things you might think about: Do you have any fillings in your teeth, metal in a bridge or crown, or a permanent retainer? Any metal in your mouth that you are sensitive to can cause really severe chapped lips. (This is part of my problem.)
The other thing is food allergies. This will vary from person to person, but is there EVER a day when your lips don't feel quite so bad? Think of what you normally eat/drink that you didn't have that day! It can range from acidic things (this bothers my daughter- tomatoes, oranges, balsamic vinegars) to things you may not suspect like gluten and corn. Since stopping corn (of all things!!) my lips have gotten somewhat better. Today I noticed they tingled just a bit when I had some grapes, so I immediately stopped eating them, but my lips are dry, dry, dry tonight from it, so I obviously had a reaction.
Do avoid chemicals on your face and lips, and petroleum products. I use plain extra virgin coconut oil on my face and lips that that has made an improvement. You can buy it in most stores now in a little tub (look at Whole Foods, or order from Vitacost.com)
I hope this helps. A food elimination seems like your best bet though, and if you have a lot of metal in your mouth, finding an integrative holistic dentist to help you get that removed safely.
M.
www.chickiepea.wordpress.com (my eating for autoimmune disease food blog)