If I understand your description right, here's what I suggest. Hoping the axle will self-center is probably asking too much. It does need to be square to the centerline of the car but you need to actually measure it. Go from the same fixed point on the chassis to the axle on both sides and ensure that the measurements are the same. Just in case of manufacturing or other defect, do the same from another pair of fixed points.
For the U-bolts, again, don't depend on weight. If they are not perfectly square to the spring then they are not at the shortest distance. The result will be slack and allow the axle to move forward and back. After the axle is squared then be certain that the U-bolts are square to the springs and tighten them to the proper torque. All should be fine then.
Bayless
Never express yourself more clearly than you can think
'48 Prefect
'67 Sprite (project)