To summarize what others have said, here (since I've done this)...
1) It is not necessary to have a NJ CCW to transport weapons in your car under FOPA rules. As long as you encase them and store them per those rules, you should be fine.
2) I concur with the I-81 route as mentioned above. It's what I've used all the time and it's a nice, scenic travel route throughout the Shenandoah valley in VA. If you live in/near Hunterdon County, it's the route to take.
a) I-78W all the way out in PA to I-81 south
b) I-81 S all the way through PA, MD, WVa, and VA. The travel time through MD is only 10 minutes and 12 miles.
c) I-77 S (from I-81 S), all the way through VA, NC, and SC (Columbia), to I-26 E.
d) I-26 E to I-95 South. Continue as you would if you had gone all the way down I-95.
3) The point of all the above (and the warnings about MD, etc), is to avoid the "Harbor Tunnels" at all costs! That is where the MTAP use those ALPR cameras and and all that data mined by the MCAC. If you got your FL CWFL some time ago (maybe 6-10 years ago), it is possible that MCAC has that information. If one of the Harbor tunnel patrols scans your plate and it flags you as a CCW holder, they *will* attempt to stop you and search your car for weapons. Refer to the case of Mr. John Fillipidis.
4) If you must travel down I-95 (i.e. you are closer to the Del. Memorial Bridge than I-81), you can take the US-301 alternate route down the Eastern Shore of MD and across the Chesapeake Bay Bridge. That will take you across US-301/50 to the Capital Beltway and on to I-95 beyond Washington DC. you don't have to touch DC.. And yes, STAY OUT OF DC WITH ANY WEAPONS! To take the US -301 route:
a) Find your way to I-95 (Del. Turnpike) however you get to it (either I-495 or Del. Memorial Bridge).
b) Take Del Tpk to Del route 1. in Christina. You'll pass right by the Cabela's in the Christina Mall.
c) Take Del 1 south All the way to Del-896 (Boyd's Corner Road).
d) DL-896 (Boyd's Corner Road) West to US-301 South. Follow US 301-south all the way down to Bay Bridge.
e) Continue US-301 S / 50 West to the Capital Beltway and then go south to I-95.
Good Luck!