That is pretty gross! But I do that with my fingernails if I'm in public and biting them. I won't spit them out, I'll just swallow them. Not if I rip the whole nail off though. That's just gross. In that case I'll pretend I'm wiping my lip and pull the nail out of my mouth. Then I rush home immediately and brush my teeth lol. jk.
I'm like you Honey in regards to dry lips being a major pet-peeve. I used to think it was the eucalyptus trees all around me when I was younger, because I'm allergic to eucalyptus. But I think it's just life.