Dad...for a second I want to try to completely forget about how YOU feel, and try to imagine how your children feel. Try to remember events from your own childhood that made you feel homesick or missing your parents, to try to help get inside their heads. Imagine what you would have felt like at their ages had you been separated from your parents, against your will.
Once you have done that, then try to imagine what the adults in your life could have done at that moment, and what you could do at this moment, to try to make things work better and more smoothly, for your children. Imagine that you were nervous or afraid of the person who was going to be separating you from your parents, imagine what it would have taken for you to become comfortable with that person.
Again, forget completely about you, forget completely about your desire to get them into NJ, forget completely about what mom has done, forget completely about everything but what would make them feel the most comfortable with you and what would make them WANT to spend time with you.
THEN, design a progressive visitation plan, that slowly integrates them into your life, and propose THAT, to the judge. Tell the judge that you understand that you are a stranger to your children and that you want them to WANT to be part of your life. Tell the judge that despite the fact that their mother behaved horribly, that you recognize that they are the ones that are important here, not you or mom, and that you want this to work for THEM. If you can do that sincerely, without falseness then the judge is going to be impressed with you and will bend over backwards to help you make that work...AND you will also be doing the best thing for your children.
You also have to recognize that the US is a whole new experience for them, particularly if their English is limited. Pulling them out of the school that they are in now, to transplant them once again, may really not be in their best interest. It also might be helpful to ask the judge to order that mom make the children available for counseling between you and the children, hopefully with a counselor who speaks both English and their primary language.