If possible, when that trailer is full, take it somewhere and get it weighed. Truck scales, gravel place, check your area to see what's close. With that information you will know how much you are towing and drive accordingly. Check and adjust your tire pressures on your truck and trailer before hitting the road. You'll probably want to be close to the maximum tire PSI on both vehicles for the best road manners.
U-haul should also be able to tell you the empty weight of the trailer. I suspect with careful packing of the trailer and the bed of your truck you will be fine. Just use common sense with regards to your towing speed (OD turned off), keep your distance between the vehicle in front of you and so forth.
Now I was curious, so I just looked up the 6x12 enclosed trailer on
U-Haul and found the trailer (empty) weighs 1,800 lbs. Max load = 2,600 lbs. GVW = 4,400 lbs. Surge brakes for the trailer. Based upon those numbers, you will be fine. Good luck with your trip and drive safely!