Oy — I’m so sorry you have to deal with such a stressful work environment! I would say to quit and find another job — since there are many “toxic” work environments and the best thing is often to leave, since they’re too hard to change. But if you’re in a training program I know it may be impossible to switch to a different one. In any event, I hope you’re not facing years of the same ugly work environment.
There’s no reason to be embarrassed for who you are! Everyone has their own challenges. Some people are very sensitive to stress and criticism, and that’s one of their challenges. It doesn’t make you better or worse than anybody else. In some Torah sources the ideal is to achieve a level in which someone can insult you to your face but it doesn’t bother you — “Let my soul be like dust to those curse me” — but few people achieve that, and it can take a lifetime.
It can be painful or embarrassing to think about past negative experiences (like crying in front of coworkers). Just not thinking about them can help. When thoughts about them do arise, don’t beat yourself up about it — instead, thank Hashem for the challenges he’s given you.
It may seem absurd to thank Him for seemingly bad things, but it’s even in the Shulchan Aruch that one should always say, whatever the merciful one does is for the best. (This approach is explained in much detail in the book Garden of Gratitude by R’ Arush.) If you think about it a bit, you can usually think of some way in which what happened was really for the best.
I also recommend spending a few minutes, maybe 10 minutes a day, of prayer in your own words each day not only thanking Hashem for everything, but also asking for strength and guidance in dealing with your job.
You might also find repeating a certain phrase in your mind (for example, “Lord of hosts, happy is the man who trusts in you” from Tehillim, as recommended in the Yerushalmi) while you’re at work. As snowbunny suggested, a few deep breaths (with your eyes closed, and focusing your thoughts on one simple thing — like the greatness of G-d or your love for G-d or His love for you) can work wonders, if done several times throughout the day.
Anyway, I hope this helps, and may Hashem help you succeed in every way!