Plenty of songs might get me close to tears when I hear them; and there's a fair few I'd be a bit nervous of trying to sing because I might not get through them.
As for sickening - I can't think of any traditional song that I'd class that way. Some performances can be, the ones that try to squeeze out all the emotions and put them on display, and over-dramatize instead of letting the song take the weight. (toadfrog's comment on Silver Dagger leaves me puzzled - maybe it's some messed up version you're thinking of. I can't remember who it was suggested it's probably about incest, which is sickening enough thought - though very possibly true - but in another sense)
In fact that's probably true of a lot of modern songs that might get labelled as sickening - it's the way they are sung that is at fault. Maybe with modern songs people find it a bit harder to think of the song divorced from the performance.
Going back a bit, and in the American context, there are all kinds of verses the Carter Family sang which would come across as sickeningly over-sentimental from many singers. But the way they sang them, they don't come across that way. Might make me feel close to tears maybe. Nothing wrong with that in a song.