A related post is here.
Are police required to record in car dashcam video for traffic tickets
in NJ, USA?
Is there any way to find out if they aren’t telling the truth?
Ask and hope you are not lied to.
Can I contact the police chief, mayor, or municipal judge?
You can contact the police chief or mayor if you can get through to them. They are not required to answer you unless you make a formal discovery request or public records request.
You cannot make ex parte contact with a judge when the other side's lawyer (in this case, the city's lawyer) is not present.
How can I defend myself in court trial if the judge always believes
highly credible police officers over defendants if I don’t have video?
You can tell your side of the story under oath with any details that makes your story believable. You are correct that the judge will usually believe the police officer and not you. So, usually you will lose.
This is one reason that most people try to plea bargain their traffic tickets, rather than going to trial.
Due process rights give you an opportunity to tell your side of the story when contesting a ticket, but it doesn't give you a right to win (even if you are actually in the right) if the judge or other trier of fact doesn't believe what you have to say.
It does not appear that you have a right to a jury trial in a traffic case in New Jersey, although this depends to some extent on the kind of violation being charged, so you are probably stuck with the beliefs of the municipal court judge about who is the more credible witness.