There are several factors which can induce trailer sway whilst towing.

One is the tongue weight. Many "rules" include the statement that a BP trailer should have 10% of its weight on the hitch. A positive weight IS necessary, but your individual towing requirements and rig may vary from other owners. The trailer will almost always sway if there is a negative weight at the ball.

The second factor is how level the trailer is sitting. If the hitch is low at the ball, more weight will be placed on the forward axle and this can cause a swaying condition. It is better if the trailer's frame is ground level or even slightly elevated, riding nose high at the hitch.

The trailer's tires and their inflation can cause poor handling. Bias ply tires are less responsive than radial ply tires and will offer less road holding performance. The differences can be especially noted with cross winds, approaching trucks and cornering loads. These causes will be more pronounced with a taller trailer, as there is more "sail area". The inflation rate is critical if the tires are run under inflated. Unlike vehicle inflation rates, most trailer tires are commonly run at the max inflation specification as noted on the tires' sidewalls.

Another difference would include the truck's and trailers' individual suspensions. Trailers with a torsion bar suspension ride much smoother than a leaf spring set up. The tires will not bounce as much on rough surfaces, and the rebound rate is greatly reduced. Fortunately more horse trailers are built with the torsion axles than those using leaf spring suspensions.

The truck's suspension can cause aslo problems. When we ordered our SD Ford, trailer towing and camper options were specified. The rear suspension included over load springs that contacted the main spring pack at something just less than a one ton load. We were hauling campers then, that ready to go, weighed just about a ton. When any curve, side wind or approaching bow wave from a truck was encountered, the camper would rock, shifting from side to side, when the helper springs were first engaged. With a heavier load this problem didn't exist, but at this specific loading, the problem was quite uncomfortable. With the installation of Timbren helper springs, this problem was completely alleviated. One of our GN trailers has a pin weight of less than a ton, and with the weight over the rear suspension instead of behind it, there is no inkling of any lack of stability. 

So I would check the tongue weight, its frame height at the ball compared with the height at the axles, the type of tires and their current inflation. I assume your truck's rear tires are running at a proper inflation as well.