Would you consider this to be too sensitive?

Yes you sound sensitive I guess it's not a bad trait. It's okay crying at movies and stuff, but there is this girl in a few of my classes and she crys at everything. She cried when she got a paper cut, when someone says something remotely offensive, when she misses her boyfriend, when her friend stops talking to her, when this boy got arrested at school... etc. It's ridiculous and everyone thinks she just crys for attention now. Don't get like that and it'll be fine.