I've am currently watching Oregairu (second season atm), I'll also recommend Plastic memories if you haven't watched it, quite emotional :x
If you just watch it without actually caring a bit about the characters, then your not watching something, same goes for normal TV shows or movies (except, animes are usually longer then movies (12 episodes x m +-20 minutes per episode (excluding opening and ending of each) > your 90 minute movie), some can make you pretty sad, but if you watch it without actually following the storyline it wouldn't be sad. It always depends on how your watching something, if you don't care what the characters are doing/living through, of course it wouldn't make you sad. When watching something and you are following the characters and then something sad happens, you might also feel sad because of it. Rating something on how emotional it is just wrong, it depends on situation you're watching it in. It might be a 10 for some, but if you don't care it would be a 1. Just because it's anime, there is no reason why it can't make you emotional. Neither is it a reason why it can't be funny (like One Punch Man, watch it if you haven't already). Plastic memories is a bit more emotional than Angel Beats (this is also why I'm recommending plastic memories over angel beats, and because I like it more personally, watch both I would say) , at least for me personally, some people say the same or less though, I felt pretty sad when I finished watching it, dem feels ;x I just finished the last episode of oregairu, I kinda like how Hachiman thinks at some points. Might be because I'm also kind of a lonely type. Might as well also recommend Gakusen Toshi Asterisk (The Asterisk War) (first 12 episode have already aired, but the other half will start in April) and Seirei Tsukai no Blade Dance (Blade Dance of the Elementalers) and Absolute Duo, these three are kinda familiar, so if you don't like one, you'll probably won't like the others. Also gochuumon and K-on for your daily dose of moe.