Lyric discussion by Clean Alligator 

Cover art for Alligator lyrics by Tegan and Sara

I think the song is indeed about making a scene and being dramatic while actually not wanting to do that.

I think it's about feeling upset about somebody who "runned around on her." This might have been somebody who "cheated" or someone who is ignoring her.

To me it seems that she felt the relationship or thing was more important and essential to her ( die without) and that the other person should not have runned around on her, but still it was not that serious that it feels justified to be so upset. no hissy fits, mind my manners I won't make a scene

But she still is very upset, so she has to deal with that and also with the fact that she is being dramatic and sensitive.

The song is a way to deal with being upset and this person treating her badly and a way to get " over you ' (it are alligator tears and the last parts of over you's)