Taxable scholarship is considered "sourced" in the student's home state. In both cases, the taxable scholarship would be reported only on the NJ return.
Taxable scholarship goes on the student's tax return, not the parent's. If the amount is less than $14,600 ($10K NJ) and that is the student's only income, they do not need to file a tax return.
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There is a tax “loop hole” available to claim an education credit, for the parents of students on scholarship. The student reports all his scholarship, up to the amount needed to claim the American Opportunity Credit (AOC), as income on his return. That way, the parents (or himself, if he is not a dependent) can claim the tuition credit on their return. They can do this because that much tuition was no longer paid by "tax free" scholarship. You cannot do this if the conditions of the grant are that it be used to pay for qualified expenses.
Using an example: Student has $10,000 in box 5 of the 1098-T and $8000 in box 1. At first glance he/she has $2000 of taxable income and nobody can claim the American opportunity credit. But if she reports $6000 as income on her return, the parents can claim $4000 of qualified expenses on their return.
Books and computers are also qualifying expenses for the AOC. So, extending the example, the student had another $1000 in expenses for those course materials, paid out of pocket. She would only need to report $5000 of taxable scholarship income, instead of $6000.
The IRS actually encourages use of this technique. From the form 1040 instructions: “You may be able to increase an education credit if the student chooses to include all or part of a Pell grant or certain other scholarships or fellowships in income. For more information, see Pub. 970, the instructions for Form 1040 and IRS.gov/EdCredit". PUB 970 even has examples of how to do the “loop hole”.
