Oil sitting on the shelf will slowly have additive packs separate from the base stock. For new oil still in containers, look in a brand new one vs a few years old, old one will have sediment in the bottom.
I set the jugs on (not in) my burr bench (vibrating tumbler). I adjust position of the jug every 15 - 20 min. Usually takes a couple hours and I can no longer see any sediment. I keep going for a while after that.
Realistically, if you can get the sediment off the bottom and in the machine it should mix it fine. But not sure how long it would take.
Oil in the machine will get contaminated with condensation, unless kept in a climate controlled environment. Unless it's realllllllyyyyy bad, it should dissipate as soon as the temps come up while in operation and expell the moisture out the vent lines. If you're really concerned, open the fill port on the diffs and inspect with a light. If condensation isn't bad there it shouldn't be anywhere.