If you had sway, you'd be dead now, or at least standing on the side of the road looking at your jackknifed rig laying on its side in the ditch.
What you're experiencing is the natural push-pull sensation that the air displaced by large vehicles causes when it comes in contact with your trailer. Your truck does not have enough "arse" to control the trailer when this happens, so the trailer moves the truck around.
Heavier spring bars are a good idea from a weight distribution perspective, but they won't change the push-pull sensation. The only thing that will change this is stiffening the horizontal plane of the joint between the truck and the trailer. Weight bars only affect the vertical plane.
Stiffening this joint will make the truck and trailer act more as one unit, like a large straight box truck, instead of a little truck towing a big trailer.
How do you do this? Sway control. Not the cheesy friction bar(s) you have now, but real integrated sway control like an Equal-I-Zer brand, or even so far as a Hensley or Propride. If you don't want to get a suitable truck, this is what you need to do.
It's called "sway control" but what it does is stiffen the hitch joint in the horizontal plane.
