We've made that trip many times and just stopped at Williamsburg on our return trip from FL. I84, I81 is by far the best route to go... Yes, it is further, but the time is much shorter.
Since I was in the area I decided to swing through Washington DC, even though it was on a Saturday morning the traffic was just awful, spent hours waiting in traffic on the beltway. It was so bad that we decided we've seen DC before and changed course to Jonestown PA, KOA. BTW: Jonestown KOA (formerly Lickdale CG) is a great stop-over point to and from NH (it's right off I81 , easy off-easy on).
I highly recommend American Heritage RV park in Williamsburg. Your Grandkids will love the Hippo (the worlds largest inflatable waterslide). I was there 2 years ago for the 4th. with 2 of my grandkids. Nothing like celebrating the nation's birthdate at the nation's birthplace at 105* with 22 thousand of your closest friends. The Hippo sure saved that trip as the pool water was also at 100* as well. It truly was an experience they'll remember.