All NJ movers must have a NJ license, not just a DOT or MC;  
Check if the mover has a right and it is not expired: NJ Division of Consumer Affairs

If you are relocating intrastate (NJ to NJ), movers must provide a written or virtual in-home estimate; but an in-person is the best way to protect you from surprises on move day. Have an estimator view all items to be moved and identify any issues that will affect the final cost. Virtual estimates should not be used for comparing companies.

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https://njmovers.com/moving-within-nj/