my advise to you: keep calm and don't worry. it will probably go away one day or another but only if you don't care and forget about it. don't let this affect your social life, nobody will notice it except from you. you don't even need lip balm, at least not in summer. in winter it might be beneficial every now and then (i advise you to use a non petroleum moisturizing cream, NO vaseline please).
just peel what's ready to peel without injuring the skin beneath and move on.