This is indeed what the problem is. NY/NJ interpret the FOPA traveling protections very narrowly against the traveler. In 2005 Greg Revell flew from Utah to Pennsylvania with a stop at La Guardia. The inbound flight was late, so he missed his connecting flight to Allentown. The airline decided to send the passengers by bus to Allentown, but he found out his checked baggage was not loaded onto the bus and he got off to find out what happened to it (it was erroneously tagged to go to Newark). While he was finding his bags the bus left without him. He ended up taking his bags and spending the night, then checked in for a flight the next morning for the remainder of his trip. Following federal law he notified the airline agents of the firearm in his baggage and they in turn notified the Port Authority police who immediately arrested him for having a gun without a license. He spent 10 days in jail, then another few months waiting for trial until police ultimately dropped the charges, and then it took him three years to get his gun back. He sued the police for not affording him the FOPA protection, but the federal trial court and the federal district court both ruled against him, and SCOTUS refused to hear his appeal. The Third Circuit (appeals ) court ruled that when he took his bags to the hotel his firearm was not longer "inaccessible" to him, so he was not protected by the FOPA.
http://archive.boston.com/news/nation/a ... igh_court/
There have been several such cases similar to this -- sometimes the traveler was able to win in the end, sometimes not -- and they all lost time, money, and guns fighting it. NY/NJ will clearly are hostile to gunowners and will trim, bend, and break the law to persecute them. Even in the federal courts the judges and prosecutors are largely drawn from the same pool as the state judges and prosecutors.
Now you are driving, not flying, but I would not chance an overnight stay in either NY or NJ with a firearm (and I sure would not leave my guns in mny car overnight in a motel parking lot), nor would I even set foot on their benighted story even just to drive across.