Inadequate tongue weight is most likely the problem, the second possibility is uneven trailer tire pressure from side to side. I assume the trailer is loaded evenly side to side, so that should not be a problem. I would also recommend backing the tractor on.
The sure way to find out the problem is to weigh the truck/ trailer in total. Pull forward and weigh only the weight on the trailer wheels, then weigh the truck by itself. Total truck/trailer weight minus the truck weight alone, minus weight on trailer tires alone equals tongue weight. You might be able to weigh the total trailer on the scale then just the tongue weight, but that takes some time, and the scale operator might not want to take that much time. Tongue weight should be a MINIMUM of 10 - 12 % of total trailer weight on the tongue, up to 15 % would be OK, not to exceed the capacity of the hitch.
A long bed truck places the trailer ball pretty far back from the axle. The greater the distance, the greater chance it will sway. If you're just using a ball hitch and no equalizer bars, it will have a tendency to sway more.
Adding weight to the truck bed might help but is a band aid approach. The truck rear tires must also be adequately and evenly inflated to prevent sway. I doubt the truck is the problem; I'd bet it's inadequate tongue weight.
Good Luck
Joe