Careless driving involves negligence on the part of the driver such that they're endangering other people. But you were even a step further, actually hitting another vehicle. That could even be considered reckless driving.

But even at the lower level of careless driving, you could've actually ended up with jail time or hundred of dollars in fines (maybe you did?), so I say, be glad you didn't hurt anyone and just own this thing.

Yes, take a defensive driving course to remove the points. They'll go away on their own in time, if your record stays clean, but I wouldn't chance that--if you don't know what happened in this case, you can't perfectly predict your future driving, either.

As for fighting it in court, I'm having a hard time imagining anyone even thinking about that--considering the entire circumstances. You have no witnesses who would vouch that you didn't hit the other car. The police officer might have statements from you which might interfere with any defense you might try to bring. And, if you lose in court, I'm not sure, but I believe you'd have court costs and possibly a higher fine and maybe that jail time that I talked about.

But if you want to take that route, you can google "NJ traffic attorney" or check locally for someone to consult with. But I'd say it's a clear waste of your time and possibly your money.