bryan wrote:
A new pneumatic closer still can't stop my screen door from slamming,
even after adjustment.

No matter how loose/tight I make the screw, I get a slamming door.
SOMETIMES the door will almost slam close, then the cylinder kicks in
and slows the door to a crawl.

Any thoughts?

I'd pull the pin attaching the cylinder to the door, then extend it and
see how it retracts. If it won't behave, you can demonstrate it where
you bought it and maybe test the replacement before you buy it.