I used to have a problem with extremely chapped lips too, but it was not TMS-related. I found that I was allergic to titanium dioxide -- it's practically in everything, including most lip balms since it protects from the sun. I stumbled across it because I was searching for how to treat my severe hand eczema. My dermatologist called it "eczema," but I now believe it was an allergic reaction that only looked like eczema. When I eliminated all products with titanium dioxide, my hands cleared up within a few weeks. Plus, my lips healed, which was a complete surprise to me since I wasn't even focused on looking how to treat that. Instead of a crack on the bottom lip, I had cracks on the corners of my mouth and my lips were dry all over (through all seasons). My dermatologist called it "Angular Cheilitis" and I think she prescribed an antifungal cream for it. I never ended up getting the cream since I was focused on the product elimination first. Now, all I do is put some pure, raw shea butter on it. In the winter, I find I have the normal type of winter dryness. I still use the shea butter and sometimes put Aquafor on top of the shea butter, if it's particularly dry and I'll be outdoors or want something heavy-duty before going to bed.
Titanium dioxide is also in most toothpastes that you can find at a grocery store or pharmacy, so I had to find a special one online. Many people have reactions to the SLS in toothpastes (sodium lauryl sulfate), but I found that for me, it was the titanium dioxide. I've been using Jason toothpaste, which has been fine.
There is one Tom's of Maine toothpaste that I could find at my grocery store that doesn't have titanium dioxide, but it would still make my skin problem act up. It is made on the same factory line as all their other toothpastes that DO have titanium dioxide. Back when I was going through all this, I actually contacted the company to find out more, and they replied that "between batches all equipment is steam cleaned and flushed out and then tested to insure there is no residue left from any previous material. It is a very thorough and supervised process.” That sounds pretty thorough. However, if you've ever used sunscreens with titanium dioxide or zinc oxide, you know how sticky and hard they are to get off your hands. I used a sunscreen spray on my daughter from Goddess Garden that had both titanium dioxide and zinc oxide. The spray left an outline of her footprints on our front steps, and it lasted through an entire New England winter! The next summer, it could still be seen. We thought it was funny to have her stand by it to see how much her feet had grown in a year. So, I don't believe that steam cleaning really gets rid of ALL the residue... (Of course, in terms of sun protection, it worked fine, and my daughter has no problems with titanium dioxide -- I only wanted to bring it up because it was interesting to me how long the imprints lasted on slate!)
Finally, I also had a few makeup items that note on the labels that they are made in a facility that uses titanium dioxide. This sounds like the way food packaging has labels that read "made in a facility that processes nuts". Based on everything I've been through, I suspect that allergies to titanium dioxide might be a lot more common that is believed.
Some other good things for lips: coconut oil and a product called Egyptian Magic (it's mostly olive oil, beeswax, honey and royal jelly). These are all ingredients that can be beneficial if it's actually a fungal issue instead, but are also just gentle and soothing in general. Raw shea butter is my favorite though.
I hope this info helps you! Good luck!!