Ok here's a crash course on how to do this. This is a very common issue and sometimes even professionals struggle with it. If you've been to enough live theater and vocal performances, you've seen people get emotionally overcome and struggle to get the job done. It happens, we're human. So how to deal with it? This is what I learned from my years making a living on the stage:
1) Rehearse. Not just any rehearse mind you, but rehearse getting yourself worked up emotionally. Reciting your story without crying to 10 people is just a big waste of time as far as this goes. You've got to recite your story and lose it emotionally in front of a few people. For actors and singers we have a regimented rehearsal process with our colleagues on a production where part of the goal is to find that edge of emotion where we can still function but give the most compelling performance we can. While not playing a scripted part, you as an interviewer need to find that same edge and the only way to find it is to practice going over it a few times.
2) Recognize your tells that you're getting to that edge and have a strategy to collect yourself. My personal one is to stop speaking and and count 2-3 breaths, then start again.
3) If you're still struggling, it's time for the big guns. You need to have a plan for something to think about that will immediately pull you from the emotional brink. What works for me and many other artists is to picture something extremely sensationally sexual, the more outlandish the better. For whatever reason, this part of the brain seems to shut down the sad emotional pathway pretty fast. Just keep upping the ante with your mental pictures. Sorry if this sounds terribly crass, but it's a tried and true technique that has saved many a performance for me personally and for many of my friends. Also useful as a guy during a sad movie if you don't want to lose it in front of your date.
In the end you may still go over the edge and cry and that's okay. Just take a breath, get a tissue, pause, and keep going.
