oh no! so sorry to hear you're having a hard time with this. i hope you're no longer licking your lips because you're right: it dries out your lips and makes them peel.
this one's important but hard for so many people to follow through: try cutting down your sugar, bread, and dairy intake. i learned that my horrible peeling lips were caused by certain foods that i had No Idea i had a reaction to. (eg: i'm partially lactose intolerant; also many fermented foods like sauerkraut and yogurt make my lips peel. most processed foods make my lips peel as well.)
also: don't put any crap on your lips except maybe some non-fancy lip balm that has No flavors, No colors or extra stuff in it, no matter how good those fancy ingredients sound. they may just irritate you even more.
finally: check with your derm to find out the Root Cause of your peeling lips. it could just be a simple fungal infection. they can fix that for you. :)