You'll have plenty of payload for the family and a decent size TT but I think you're pushing it with 6K dry.
The E350 shows a max tow rating of 7500lbs. That number is for an empty van apart from 150lbs for a driver. Any weight added to the van reduces your tow rating by the same. If you add 1k of passengers and gear to the van then your max TT weight is now 6500lbs LOADED. A 6k dry TT will easily weigh 7500+lbs once options are added and it's packed with gear for 9 people. That's asking a lot from the 5.4l engine, especially if you're going to be in the mountains.
Another issue is the GCWR that you list. If you estimate that the van weighs 7k and the GCWR is 13k then your max trailer in that case is 6K LOADED not dry.
Best place to start is to load the family in the van and head to the nearest truck scale and get some accurate weights.