Dukeboy has it right... the leaf suspension is going to move sideways quite a bit during cornering. That is at least half of the problem, if not most of it. How much clearance is there between the tire and body now, when sitting still? Push repeatedly sideways on the rear bumper at a frequency that matches the natural sideways frequency of the rear (like pushing a swing higher and higher) and you will get some idea of how much it moves. An anti-sway bar is not going to fix ANY of this movement.