How do I interrupt a negative thought process?

I am trying to change how I think about and react to certain situations. For example, yesterday I had an appointment at 1:30. The person I was supposed to meet showed up at my workplace at 12:50 and said the appointment was at 1. I was not going to be able to get there until 1:15. I had in writing that the appointment was, in fact, at 1:30, but I still found it stressful that I would now be “late.” (And had I been the one in the wrong, I would have been frantic.) Being stressed by this is unhelpful. I chose to simply not think about it, deliberately pushing the thought away each time it arose, and distracting myself with other thoughts, but that doesn’t “process” the emotion.

If I thought about it, the stress continued to build. Do I process the same emotion repetitively? Is not thinking about it an appropriate choice? This is an almost daily occurrence for me…anything that happens where I find myself in the wrong makes me anxious, but thinking of it that way causes me to spin unhelpfully. When I realize that an emotion is needless and unhelpful (stress, anxiety), I want to learn how to move past it, but I don’t know how. Again, if I process it, it recurs multiple times. Thank you for your advice.