I tow nearly the same weight as what you are looking to tow. UVW 6200lbs, hitch weight 680 lbs. When loaded I am at about 7200 lbs give or take a couple hundred pounds depending on the trip. always tow without water, have 2 older kids that come along.
My 2009 Tundra Crewmax tows this trailer extremely well for the last 8 years. Based on what I am towing you should be fine. I would not buy a new truck just to tow that trailer. The Tundra is a very capable tow vehicle. As others have stated payload will be the first issue but pack sensibly and you should be fine.