Self help evictions are illegal in NJ. I would pay the rent by certified mail, RRR to the address in the lease. She can not require you to pay rent early and if you pay the rent in a timely fashion, she cannot require a late penalty. With the certified receipt, keep a copy of the check a cover letter and any other enclosures. This way you have proof of timely payment.
Mr. Console is correct. Assuming the utilities are collectable as additional rent, you may want to consider bringing an eviction action for utility bills. That will either prompt payment or get the out.
This is not a defense to the violation. I agree with counsel that you need to contact the court clerk, them know you plead not guilty andthey will reschedule. You can do this your self or retain an attorney
The first thing you need to do is read the lease. In there should a provision discussing how to send notice to the LL. Typically one must notify the LL in writing, served either personally or by Certified Mail, Return Receipt Requested. You must send notice by the specified method listing the issues and giving the LL a reasonable chance to cure. Assuming nothing happens, you have several options. You can break the lease for breach of the warranty of habitability, repair the issue yourself...