There are indeed very poor schools in New Jersey, but it all depends on where exactly in NJ you can afford to life.
If you go to a good school district, as mentioned in my previous post, I would not worry to much. Save the monney for the private school, and use it to buy/rent you a house in a good school district.
Certainly you need to ask the school some information about what they would to for your kid that needs some extra.

At first, when they mentioned relocation to nj to us, I was certain that only a private school would be good enough for our son. He's a 13 year old 9th grader in a renownd Jesuit school in Belgium. But after I did some research and had contact with other Belgians who lived/had lived in the neighborhood, I now am convinced that we don't need a private school. But that we pay some extra for the good school district.
I can not help you with recommandations about an elementary school, because I only took a look at middle and high schools.

(and sorry about my English, I'm not used to speak or write it, not yet)