I think staying calm is very important, but there's a fine line between calm and just downright slow. I think the best condition is sort of an urgent calm, where you know your goals and what you're going to do in the stage, and you know you need to go fast, but the urgency doesn't turn to a panic. Panic isn't going to allow you to pay attention to what's going on in the stage. It's tough to find that happy medium, but when you do, it really does feel markedly different. I kind of think of it like those picture books, the ones where it's just a weird abstract picture, but then you start to change your focal plane and suddenly, it's a snowboarding kangaroo. I've been "in the zone" during shooting maybe once or twice, and I'm trying to figure out how to cue it to happen, but it really is cool when it does.
All that being said, I think the key is where you're focusing. Normally, when what you described happens to me, it's because I'm focusing on the entire stage at once. I find it's better to focus on specific goals, which is why the runthrough is so important. Break it down into parts. When you are waiting for the buzzer, it may be best to clear everything out of your mind except for where you're going to do first. If your starting position is hands at sides facing downrange, you should have your draw so ingrained that all your focus is on the first target. Once that target is down, focus on the next, and so on. When starting uprange, just focus on getting turned around and the presentation to the target. Stuff like this helps to slow you down enough to make more focused shots.