Most of us have learned to manage this effect to some degree, some more successfully than others. Every officer knows someone that is remarkably cool under pressure, and someone that just can't seem to hold themselves together. Generally we admire the former, and pity the latter. The fact is that we all have elements of each type--some of us just react differently.
When body alarm response kicks in during a vehicle pursuit, we fall into a condition known as "pursuit fixation." We suffer all the typical responses to body alarm stress, within the context of an intense, focused mission. We are after the fleeing bad guy and our mission in life becomes his apprehension. We are focused on our goal--to catch him!
There are a couple of common by-products of this situation. First, it's not uncommon for officers in these circumstances to get angry. There are lots of theories as to why this happens, and my friends in the police psychology field probably have medical terminology for classifying it, but ,in the most simple terms, I think a lot of officers just get p***ed off that the guy is running. Whether the officer sees it as a challenge to authority, or is unhappy at having to put himself or herself at risk by driving fast, or maybe just doesn't want to get their freshly washed cruiser muddy, they get angry. Anger colors judgment, and in a police pursuit, we cannot afford to have our judgment impaired by any extraneous emotion.
The second by-product is single-mindedness. That, in and of itself, is not necessarily bad. The problem occurs when that single-mindedness is focused solely on apprehension of the violator. When this occurs, sometimes officers lose sight of their primary mission, which is to preserve public safety. After all, the very reason the officer is trying to stop this suspect is because the bad guy has broken the law. Whether it's a traffic offense or something more serious, the law was most likely put in place to preserve public safety in some regard. The bad guy breaks it, thereby endangering public safety, and the officer tries to apprehend him in order to allow the justice system to control the deviant behavior. The bad guy flees from this effort at control, and the officer gives chase in order to re-assert control.
The very nature of police pursuit is to enhance public safety. In order to pursue, we--by definition--put public safety at risk. It is the job of the officer to balance the need for the apprehension (and therefore the enhanced public safety that will result from apprehension) against the risk to public safety caused by the pursuit itself. Under the stress of the pursuit, in that moment when body alarm response is challenging the officer's ability to rationally and analytically analyze options and select courses of action, he or she cannot afford the luxury of anger.
What can you do about it?
You cannot avoid body alarm response. It's natural and normal. All beings experience it. In fact, you don't want to avoid it--it's what makes you sharp and aware, it provides your sixth sense, your "survival awareness," and your enhanced ability to hit harder, run faster, and scream louder. Body alarm response is a good thing. It's the outward manifestations of body alarm response, and the way that they can interfere with your other abilities, that create the problem. These manifestations can usually be managed, and the more you can learn to control them, the calmer you will be under stress, and the greater will be your ability to think clearly, and to survive.
First of all, and probably the most important thing, is to be aware of the natural nature of body alarm response. Read about it, study it, learn all you can. Such books as An Intimate History of Killing, and Fear: A Cultural History, both by Joanna Bourke; The Gift of Fear, by Gavin De Becker; and Emotional Intelligence by Daniel Goleman, are good places to start.
Talk about it with your trainers and your fellow coppers. This is just a fancier version of the "What If...?" game.
Seek out formal study in fear and anger management. Learn calming techniques. Practice a martial art. Consider taking any of the excellent classes offered by Brian Willis of Winning Mind Training, or others geared toward developing your ability to enhance your mind's control over your physical reactions.