Do you actually feel like you’re sensing a weight shift? If that’s true, than it would strongly suggest that OK4450’s theory of an eccentrically worn joint component is correct. Include in the definition of “jont component” an inside CV joint.
The inside CV joint allows the axle to change length by allowing a three-pronged bearinged center to rotate with tortional load applied while also sliding within a tri=slotted housing. If there’s a detent worn into the slotted housing, when you steer and the triad assembly (my term) is pulled out of the detent, that could be affecting the steering.
Caveat: my theory in a wild one, but if true should be verifiable. You should be able to feel the detent on the ground at a dead stop. or with the suspension unloaded on a 4-point lift.
One other possibility that ooccurred to me is a malfunctioning spool valve in the steering rack. Again, that should be verifiable.