There are certainly exceptions but the general experience I have seen here is that if you move from an upfront taxation state (whether just on total amount of lease payments like NJ or on total value like VA) you generally get screwed.
A quick Google search doesn’t turn up any provision in NJ for getting the tax refunded. The DMV mentions the form you cite to but in relation to a canceled lease, not moving out of state. And this makes sense from a basic political perspective. Why would a state want to refund money to someone moving out of state. You aren’t a voter there anymore.
For purchased vehicles it is simple, almost every state honors sales tax paid in the previous state (note the state you left doesn’t refund the sales tax). Not sure why a similar system doesn’t exist for leases.