You can look down into the channel the carriage rollers run in while prying against the top carriage rollers (pry tool between roller and mast channel) to shove the carriage over so you can gauge how much slop there is, and why. Alternate the pry tool from one side of carriage to the other to see he full range of free travel.
When you say " the main cylinder bounces up and down before settling", do you mean the entire cylinder assembly or just the ram/rod part that moves in and out of the cylinder.
I'm pretty sure you mean just the ram.
When a lift cylinder ram bounces up and down when lowering loaded, it indicates there is air trapped within the cylinder.
Air in a lift cylinder may be due to an oil leak on the lift cylinder circuit that allows oil to drain out, and air to enter around the seals at the top of the cylinder while the truck is parked and not in use. So, if the lift circuit has any oil leaks that you have simply been ignoring, that could be the source of air infiltration.
Another way air can become present in hydraulics is if the hydraulic oil is very old and has not been changed in a long time. The oil loses a good deal of its anti-foam additive package and air can become entrained in the oil while the truck is in use.
Later when the truck is parked, the entrained air can come out of the oil and collect in small pockets above the oil.
Trapped air in a lift cylinder can usually be expelled by tilting the mast back as far as it will and then lower the forks carriage completely a few times (raising to to full height and then lowering completely each time).

I will also grant you that mast configurations where there is a single center lift cylinder with a "cross head" chain roller assembly on the ram can be prone to the carriage having much more "see saw" motion than masts having twin lift cylinders with a wider spacing of the carriage chains.