If you have already paid the fine, it may be possible to withdraw your guilty plea and get the ticket reduced to a non-moving violation. This is more difficult if too much time has passed or if you were represented by an attorney when you pleaded guilty.
Like the other attorney said, there is a very good chance to get your old tickets removed. You will almost certainly need an attorney to help you, however. If you plead guilty and paid the fine (without an attorney) way back when, an attorney can file a 'Motion to Set Aside a Plea of Guilty' indicating that you were not represented and didn't understand the points system and its consequences. In my experience, most towns in this state are broke, and they'll gladly remove your old tickets and points from your record in exchange for some cold cash. If you can't find an attorney in your area, give me a call and I can help you out.