Well right off the bat, towing is never going to be like a Sunday ride to church in the family minivan. You've got a tow vehicle with a stiff suspension meant to carry weight, not ride like a cloud, and you've got a massive trailer behind you with even less attention paid to ride quality in the suspension design.
Bucking and bouncing can indicate that you've got too much weight on the truck and/or the weight is too far back. I assume you have a shortbed truck. Do you have anything like a Sidewinder hitch that places the actual pin of the trailer near the rear of the truck bed? Are you running a slider hitch in the rear position? The factory location of the pin should, ideally, be within a few inches of the truck axle.
Weight wise, unless you are all over 250lbs, or carry massive amounts of heavy items in the truck bed, you should be okay. The factory pin weight is just shy of 2000lbs on that trailer. 3000lbs, even 3500lbs, between pin, passengers, and stuff, is no problem for that truck.
How does it sit? Level? Squatting? If squatting you might want to add Timbrens or airbags to level it out.
Shocks never hurt, but I don't think they'll necessarily help in this case. That truck is 4 years old and depending on the mileage the factory shocks just might have bit it already.
If the truck is not swaying a sway bar won't make any difference.
Putting 10-ply tires on half ton trucks since aught-four.
