The softening is a feature, not a bug! The skin that comes off and rubs off was already dead.
Some of it, anyway. However I think some not-useless skin tends to come off with it; what's underneath tends to be somewhat tender, as though I peeled off a scab.
Oops, sorry, I failed to spell things out here.
I suggest not rubbing off the skin when it's softened. Some might fall off on its own, and that's fine, but don't fiddle with it. The analogy with a scab is a good one: a scab also gets softened and easier to rub or peel off when it's wet, and a scab is also not alive, but the skin underneath is not fully healed yet. Removing a scab or the flaky skin from your lips before they are ready to fall off naturally doesn't help the skin heal, and it hurts. As the old doctor joke goes, "Don't do that, then."...
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?