Sometimes I catch myself trying to tell stories, but stopping when someone else starts talking at the same time. At that point, I consciously try to fight the urge to keep my story from staying front of mind. If I don't, I end up focusing most of my attention on waiting for a lull in the conversation instead of actively listening and contributing, which is a waste of everyone's time.
One approach that's worked well is to write myself a note so I don't forget what I wanted say. This way I can move on. If I'm talking with people I see often, I don't worry about getting to share my story because I know I'll have another opportunity. And if I'm talking with people that I don't see often, chances are that it doesn't matter that the moment passed.