As nice as that trip could be, this sounds like a guaranteed way to make it memorable (not the good kind). What you have outlined is what in younger days we called a bonzi run. My son went to school in RI for a while. (Only 960 miles) We could do that in two days in a mini-van. It would be two real hard days in our coach and our is a comfortable ride.
I have two suggestions:
For this run, ship as much as you can by what ever means and the two of you fly.
For a future run and better time - like spring - pull the youngers out of school (you can bring the lesson plans and books) and do the run as six days each way on the old Route 66. This will be a lesson those kids will never forget.
Matt
