I think you definitely have a chance, and you need to start with an on-time appeal of the denial of extended benefits.  If unemployment won't accept the appeal, or if they deny it, you should hire a lawyer and take it to court;  it will probably have to be taken to the Appellate Division of Superior Court, and this is most certainly not for the faint of heart, even among admitted attorneys.

Due process requires that you have notice of what is at stake, before you give up your rights;  if you reasonably didn't know that you were waiving the extended benefits, on top of the 5 weeks, by not appealing the 5 weeks, my understanding of New Jersey law is that you can't be denied the opportunity for a hearing on this.