I've used my 4Runner to move twice. First time was from Sedona, AZ to Scottsdale (2 hours) and I used the 5x8 U-hual trailer. I actually destroyed the rear shocks. I must have had too much stuff in the trailer and truck; both were loaded.
Just this past December I moved from Scottsdale, AZ to Seattle WA. This time I opted for the 6x12 trailer because of the double axle. What a difference! The weight of all my crap was supported more by the trailer then by the 4Runner. So if you can I would recommend a double axle trailer.
Here is my post about moving if you want to check pictures/read through it
http://www.toyota-4runner.org/3rd-gen-t4rs/158046-6x12-trailer-phx-seattle-12-28-a-2.html
I'm sure a lot of people will weigh in on this, but I only turned the OD off when I had to. If I was on flat ground then the 4Runner pulled fine in OD. Just plan a route with the least amount of mountains. It does not like to pull a trailer up hill. I had to drive on the shoulder a few times with the big tractor trailers haha.
Also, I did fine without the tundra upgrade. I will be doing it for my next move in November but just keep a safe distance. At every stop I check all the tires, fluids, etc. Keep us posted, its always interesting seeing people haul trailers with these long distances.