The softening is a feature, not a bug! The skin that comes off and rubs off was already dead.
When healthy, your lips - and indeed your whole skin - have oils in the outer layer, to keep the skin flexible and to act as a mechanical barrier (keeping the outside out and the inside in - in particular, water). When this oil isn't there (whether your skin doesn't produce enough, or it is removed by something), your skin dries out, and starts to flake. It typically gets very sore too, and prone to infections.
Drinking more might help, but frankly I doubt it unless you are chronically dehydrated and have other symptoms. It's far more likely a dehydration problem local to your lips.
I recommend training yourself to develop a lip salve/ChapStick habit - particularly before you go outside (when you're most at risk from lip dehydration), and before bedtime (when it has a good chance of staying on undisturbed). I also recommend training yourself out of any lip-licking habit you may have. Licking your lips a lot tends to strip the oil from them. This can become a vicious circle if you are licking your lips because they are sore.
Lip salve works because it (a) adds oil to the skin, making it more flexible and thus less prone to painful cracking; (b) provides a mechanical barrier to loss of moisture; and (c) provides a cue to help discourage you from licking your lips.
Some ideas for developing a lip salve habit: Buy lots of sticks and put them all over the place, so you are never far away from one. Put one in your outside coat pocket so you notice it when you go out. Put one on the shelf by the door where you pick up your keys before you go out. Develop a mental link between putting on a hat (to protect your head from the cold) with putting on lip salve (to protect your lips). Put lip salve in places that you can't avoid: balanced on the outside door handle, on seats you use the whole time, inside your hat or gloves, etc. If you are avoiding lip salve because it tastes horrible, experiment with flavours until you find one you like. (If you are licking your lips more because you like the taste of the lip salve, or find it less aversive, you may need to make a compromise.) Set a reminder on your phone every hour (or at pseudorandom intervals) to remind you to put on lipsalve. Log how many times a day you remember to put it on and monitor yourself on Beeminder.
I've heard that if you use lip balm too often, your lips lose their ability to lubricate themselves. I don't know if this is true or not, only that I can't go more than a few hours without applying lip balm without experiencing the same symptoms CronoDAS describes, and this has been the case for as long as I can remember.
With Alicorn's permission, I'm resurrecting this thread.
I'll start off with one of my own: What kinds of exercise can I do at home (I do have 5- and 20-pound weights), and what are good ways to get motivation to do so regularly?