I asked a similar question a few years back, of course my story was a bit different. Today, I still get highly emotional/hypersensitive, however I feel way better than I did before. My psychologist said about the same thing to me, except she phrased it like "you make it seem really bad that you're emotional/sensitive". At first, I didn't take her words very well, but she was right. I wasn't accepting who I was, and if some people can't handle my sensitivity, they don't deserve to be my friends. I did more research on being a big cry baby all the time, and believe it or not, there's TONS of people like us. After I accepted who I was, I started noticing others similar. They were the most gentlest, caring, and had the brightest smile.
However, too much sensitivity can really cause a drain in ones energy. So I'll give you some tips on how I became slightly less sensitive and hopefully it would work to the best of your luck. The biggest link of depression is negative thinking. Think positively, even if things are falling apart. Try starting small first, like if someone cancelled on you, instead of "no one likes me" think "now I have more time for myself".
Once you got a little power over your thinking/mindset, challenge yourself to desensitize from minor details that you Would usually care about. I'm not saying, stop caring about everything, but pick which things to cry over, and which to not. If something on tv made you upset, that's completely normal, you can cry about those things because it really doesn't matter if you did or not. The only problem is people who don't accept other people crying from something that made them emotional. They simply don't understand, and that's not something you need to stress about. The world is not going to end. This is usually easier once you feel more in control of your mind, or rather how you perceive things. For instance, if you lost a game and your partner yelled at you, don't worry about it. Justify your actions, instead of criticizing. Accept if things are a mistake, and learn from it, rather than dwelling on it. It's better to move on knowing what you did wrong so you can improve on it, then to move on feeling guilty and weeping about it.
Whenever you start to feel upset/emotional, analyze your feelings. What made you feel this way? What exactly happened? Will things get better if you cried? Are you going to be fine? Breath, smile, and most importantly, find someone you can trust to express openly. A lot of times, I found myself bottled up with emotions, and thus believe it to be one of the biggest reasons why I would cry easily at anything. Your psychologists is a great person to go to in times of rant. Trust that being emotional and hyper sensitive is nothing to worry about. If people can't accept that part of you, then I feel deeply sorry for them.