I shoot both small bore benchrest and bullseye pistol. I have talked with a number of psychological professionals at length regarding the proper mindset for shooting.
In pistol, I find there is a time when you can get 'in the zone'. The zone is recognizable by the fact that you are not distracted by anything else. The fact someone might be watching you doesn't matter. You can't hear anything but the operation and firing of your own gun; all others on the range are blocked out. Your concentration is solely and completely 'in the game' and you are saying nothing to yourself; your head and thoughts are completely quiet and engrossed in what is happening. Sometimes you can see things happen in slow motion. When you start to notice the cycling of the slide, or the flick of the brass out of your gun, that's 'the zone'.
I am not able to just enter 'the zone' at will. I have to have it happen to me on a good day. But this is fairly common; most of us will experience some level of anxiety when we do a performance sport. Anxiety can be good, it can help you perform to a higher level than you practiced at. However, too much anxiety will lead to rapid performance degradation.
Therefore the question is: how do we manage performance anxiety?
When I shoot slow-fire bullseye, I do not like to spot my shots, except maybe the first one or two. I feel that when I see I just shot a '10', it makes me more nervous for my next shot. However, since we are talking about action shooting, I'll skip how I feel in slow-fire or benchrest.
In rapid fire, before a string, I will close my eyes, inhale slowly and deeply through pursed lips (you want some resistance to breathing in), and go to a 'happy place'. Yeah that sounds silly, but you just think of a setting or environment that is especially relaxing for you. You know, your favorite easy chair or the beach or whatever. This actually will slow down your heart rate if you do it right. It's the combination of breathing, plus the happy place that results in a change in the body. If you get a simple pulse meter for yourself, you can practice this at home: biofeedback.
The mental aspects of preparation in shooting are often overlooked. Consider talking to a sports psychologist for a session or two. Often times it can be very productive.