Why does it sway?

[Blue'72]

I think I might have an idea what's causing this, but I want to get everyone's opinion. When I put any weight in the back of my truck and then get up to speed, any movement of the steering wheel will cause the back end to sway from side to side. The more weight and the faster speed causes it to sway really bad. It doesn't take a whole lot of weight, but it's downright scary when you load it up with firewood. Saturday I hauled two yards of bark in the back and pulled my trailer with three yards in it. The trailer was rock steady, but the rear of the truck still wanted to move from side to side.

I don't remember having any problems with my 72 half ton long bed I had a few years back. Everything's nice and tight under there, but I'm wondering if the overload leaf springs might be throwing it off somehow. The truck has coil springs and then some aftermarket leaf type springs that hold it up after the rear comes down a little. These overload springs only have one mount and the other end of the leaf sets under the frame to support it. (I wish I had pictures..)

Any other ideas???