Can't tell you if New Jersey does require any such thing but amusement rides are very strongly regulated in many states. The mishap may have been required to be recorded in a log or such. I would call the state offices and find out who regulates such amusement and ask if it is required. This may get you to the proof that there was a problem at the park.
Now to your injuries: without making a report, you will be hard pressed proving you were on the ride and your injuries were caused by the ride mishap. I would make a report at this time as time is against you already and delaying will only make it worse. If they refuse to take a report, get names, postiion of employment and times to document this.
If you have any proof you were at the park that day (picures of you being nutso or whatever) keep them around. Anything you can use to provide proof you were there is important. Ticket stubs are good. Hotel stay in the area, although would not prove you were at the park but with other things it can help.
If they take a report, it still does not mean they accept responsibility.