This is an OT car also, but you can watch the relationship of gas tank to spring as it's getting wider or closer through turns. https://www.youtube.com/watch?v=MGxea8ADE18 One other thing to consider when talking lateral movement in a leaf spring car is the arc of the spring. The flatter the spring, the less lateral movement it will have. The more arc it has, the more it will affect the lateral movement.
keep in mind the car in the second video has many many more stabilization aids than the op's truck. you would expect to see movement like the first video
My 47 cadillac with rear leafs came with a panhard bar factory, but I know better so I tossed it and switched to canted tube shocks to replace the old knee action style. So far so good with about 1" clearance but I am not autocrossing the thing.