If it touches you so much I can imagine it makes it hard to listen to, don't put yourself in unnecessary sad situations if you don't have to. If you don't want to feel like that just skip the song. there will be moments when you're OK with tearing up, you can always listen to the song at those times.
It touches me also, that's why I like it so much, luckily for me it doesn't have the same effect on me. (I had it one time a little bit though, but it felt Ok at the that moment)
On the other hand: it is quite special that some music can touch you so much. It says something about the song, but also about your capability to sense stuff and putting yourself in other people's situations (and maybe it reminds you of something you went through in your life). But not all beautiful things are meant for any moment, sometimes it's even better when it's a rare thing.
