Welome to OCDOP and welcome to Virginia. Your NJ dollars are welcome here also.
I do not know how the state police handle things in NJ, but the VSP do not answer legal questions (except that you are or are not free to go). It's pretty much a self-protection thing, as well as professionalism to only deal with what is clearly their area of responsibility.
It is a long-standing truth that one should not ask cops anything about the law, and certainly not about firearm laws. I am a lawyer (one educated in the law) but am not an attorney (a member of the guild allowed to argue with a judge without immediately getting jailed for contempt), so I can offer you a few legal opinions:
on hip in car is concealed - Nope! Not even close.
in glove box is concealed - Yup! The nice thing is that in Virginia you do not need a state permissio slip to carry concealed in the glove box (so long as it is closed) or any other secured (does not need to be locked) container in the vehicle.
driving past a school within 1000 ft big trouble - federal law says you must have a state permission slip to do so without violating the prohibition. Generally speaking VSP does not enforce federal law. What they might do is, if arresting you on a state charge, refer you to the US Attorney for prosecution in federal court on federal charges that the USA brings against you. Many years ago there was "Operation Exile" in which the local cops referred as many firearms cases as possible to the USA to get the harsher federal sentence imposed. It's not nice, but it did keep some BGs off Virginia's streets a bit longer than they might have been if convicted in state court.
The next time you are going to be in town post it and suggst a time & place for a meet&greet. Some of us will be sure to bring the VCDL carry info cards so you can get a copy.