After paying close attention to the noise, I found that it was coming
from the diverter, not the shower-head or the cartridge.
I switched the diverter from another shower in the house and it did not scream in use. When I put the screamer on the other shower, it did.
Solution: buy a new diverter and the problem is gone. Thank you, everyone who had a suggestion for me to try.
By the way, I also learned something in the process. I did not know there
are two types of diverters, the screw-on type and the o-ring press-on type. Mine happened to be the latter. Most stores don't seem to carry that kind, but Home Depot does. The package even comes with an allan wrench, but does not come with any silicon "grease". It should, though.