Yep ~ I know exactly what you're talking about. I'm the Queen of the criers.
I don't have any "tips" -- The only thing I did was just FORCE myself not to cry. That's not to say that I didn't "well up" with tears in my eyes, but I didn't cry. LOL I brought some tissue with me, just in case. LOL But I was determined NOT to cry, because the issue was important to me, and I wanted to see it through.
It was similar to yours -- it was a Union issue and I had to talk to the Union steward about my boss, who had not given me ANY training when I first was hired on, yet expected me to perform like everyone else on the line.

Then I'd find her in the breakroom, taking naps all day.
So this effected everyone plus the boss who would most likely take her punishment out on me.
I got thru it okay. It wasn't fun, but I got thru it. I was a nervous wreck the rest of the day, but I didn't cry. I waiting til I got home that night, and then cried! God bless and good luck! Let us know how you do. HUgs, Lee