Expressively the addresser is constructed as begging, strong yet submissive, desirous, desperate.
Conatively the addressee is constructed as stubborn, unforgiving, an object of desire.
Phatically the relationship between the addresser and addressee is constructed as dysfunctional, rocky, loving, romantic.
Referentially, the song leaves out some aspects of the relationship between the addresser and addressee to assist the receiver in an identification with the position of the addresser (phenomenological semiotics?), but also gives enough information to help us understand the phatics of the relationship between the addresser and addressee.
Metalingually this song doesn’t explicitly state any facts of the relationship or situation of the addresser and addressee to leave the song up to interpretation of the listener.
Ne Me Quitte Pas is formally a song. It’s words vary in meaning and connotation verse to verse, but not so that the story of the addresser and addressee loses cohesion. The chorus remains the same to emphasize the begging of the addresser for the addressee not to leave.
I have no idea if any of this is right, but I thought I would give it all a try.