I've never used a Littleton, but have a fair understanding of how dropping shot works.
As 2-p said, the shot passing through a vibrating seive causes it to form more or less uniform size drops. As the drops fall, they cool in the air. How fast (how far they fall before solidification) they cool depends on drop separation temperature, drop size, air temperature and humidity. Surface tension of the drop forms it into a relatively round sphere. It will solidify "round" if it cools enough before hitting the water. I suspect the oil additive impacts either the cooling rate or the surface tension of the drops; I'm not sure how it works.
For hardening to work as stated, the impact temperature of the drops and coolant temperature would have to be controlled.
