Careless Driving: Careless Driving in Middlesex County, New Jersey

My question involves a careless driving ticket from the State of: New Jersey
and I would like to know my chances of fighting off the two points I will recieve if I plead guilty.

My info in short is:
18 yrs old
Male
driving less then two months (will not help in court, but will help for those evaluating my situation on this forum)

So heres what happened:

I was driving down a main road and was approaching a light which was twenty yards away. I looked at my right side view mirror to see if I could change lanes because i didn't want to turn left but their was a car. I looked back in front of me and all was going well until I was less then 20 feet from the light and it turned red and the car in front of me stopped suddenly and I tried to stop but hit the car in front of me. The lady I hit turns out to be a school administrator from my local high school and she immediately called the police. There was no damage to her car and she was fine although she said later her arm was hurt but nonetheless the officer asked me what happened and I recounted everything and he took interest when I said I looked at the mirror and he wrote up a ticket for careless driving on my behalf even though my story was not complete at that point but he marked me with fault in the accident.

I admit I probably should of left that part out but I felt it best to recount the whole event from beginning to end but unfortunately the officer heard "looking at mirror" and ran with it. I really want to fight this in court and plead not guilty to plead my case and tell the event again and also say how the officer only took interest in one part of the story. Do you think I have a good chance? Is it possible to avoid those 2 points being added to my license? Please advise as the best thing to do because I am willing to do anything to avoid those two points (because my insurance will double) and I am lost and scared but want the truth to come out. Thankyou