Well, if it's only for the tears - no problem: Just let them run wherever they want as long as they are in your eyes when you're feeling emotions (no joke). Of course, when your voice breaks and you can't control your breath any more (i.e. you are really weeping) that won't be good if you are trying to sing instead of it...
I don't feel able to comment anything of the above, I can only tell you my own impression: There are songs I just can't do first, because of this very problem. The first ones I encountered were "Bringing Mary Home" and "A Daisy A Day" MANY years ago. (I don't do them any more because I am disappointed of my performing when I try them today, except for some special situations which hardly ever occur, and then I'm not thinking of doing these songs.) Recently I ran into "Georgia Lee" by Tom Waits: Q§$%&! - nothing has changed since then. Well, I know the solution that works for me: Repeating it often enough will calm me down, and practice won't hurt the technical quality at all. But that will also 'kill' the song for me: I'll get used to it, my emotions will be gone, and the song will bore me and therefore my audience too.
That's why I'm literally consuming songs: I practice them until I can do them without crying, then I do them as long as I can still feel them, and then ... *sigh* ... no, I don't forget them.