I had this problem for 20 years.
My problem sounds similar to most on the board. My mouth was just dried out from a different medication (cold pills). And mouth breathing can also dry out the mouth.
When I first came down with this problem, my lip skin flipped up dead one day shaped like shark's teeth. It all happened in one day! As years went on, my flipped up lip skin turned normal color (clear/whitish dead skin). But in the early years, my dead lip skin was this light yellowish brown color (I'm trying to recall the color).
I have the peeling cycle you all have. Next, comes the raw skin. Then the new skin grows too fast and too much. It becomes hard and peels off in sections.
I recall pulling the skin hanging off a triangular area on my upper lip when I was a child. I can't say that I was picking at my lip. It might have just been that I had a hang lip like a hang nail. Myabe that area was more prone to infection.
If you a female like me, you probably get more of that white line on vermillion border since it often occurs when your lip is moist. Lipsticks make your lip moist. I wore moisturizing lipstick since my lips were so chapped. I had to keep wiping that white line off my lips so it was hard to be out all day unless I could get to a bathroom. I do not take allergy or cold pills on a daily basis like I did when I was a teenager, but I sometimes take them. When I take these pills, the white line got worse (maybe it dried the plaque or whatever in my mouth so that it became more thick.
It has been hell.