Well, it's quite old, but 'Ne me quitte pas' (by Jacques Brel) is probably the most yearning French song possible. Or 'Et si tu n'existais pas' (by Joe Dassin) but honestly I don't listen to it
There are also (way more modern and less dramatic) 'Amour plastique' and 'Roi' (by Videoclub), 'Ma meilleure ennemie' (by Stromae), 'Melodrama' (by Théodo......
I can't wait for Paula and Vincent to be in loveeeeee