OK - The damper plate has been replaced with the same Centa model DS. The old one was completely thrashed.
I aligned the prop shaft to within .002" gap at the gearbox coupling coupling.
I started her up and took it for a test run. It idled along in gear very quietly and smoothly, better than it ever had. I slowly took it up to 2000 rpm, then to 2400 rpm . It was generally quiet, but I could hear a slight harmonic vibration which pulsed every second or so. Slowing the revs to 1800 slowed the pulsing.
After 30 minutes, I dropped back down to an idle, and listened. The harmonic pulsing stopped at under 1500 rpm, but I now had a very slight rattle from the damper plate again.
The harmonic pulsing has always been there since I owned the boat. The sound isn't obvious but it is getting slightly worse.
I am now thinking the root cause of this may be a bad stern tube.
My prop shaft is only about 3 feet long. I'm guessing the stern tube is only about 8-12" long. (no strut, cutlass bearing etc).
How much up/down or side to side play should I have at my prop shaft flange?
A bad stern tube? Can you elaborate? Do you mean shaft bearing? If it's getting worse it could very well be a worn bearing, that usually creates a hum or rattle. Clearance between the bearing and the shaft is established in the ABYC standards, for a 1.5" shaft, for instance, it is 0.004" - 0.009". This can be measured with a feeler gauge.
You mentioned "How much up/down or side to side play should I have at my prop shaft flange?" Do you mean when it is disconnected from the transmission? If so, that's entirely a function of the stuffing box arrangement, which limits shaft motion to a degree, and the gap between it and the flange. For a short shaft like yours, not much.
It is possible the shaft is making contact with the stern tube, while still being properly aligned, I've seen this several times. In most cases it will generate a noise and/or vibration, as well as wearing away the tube. You'd need to disconnect the stuffing box and look at the shaft where it exits the tube to know if this is occurring. Ideally that's best done while hauled, but it could be done in the water, in a pinch, if you aren't too squeamish about some water coming in. You might also be able to feel it making contact with the flange disconnected, but that may be difficult if you have a stiff stuffing box hose. If it's a dripless seal and a more pliable hose then that's easier to make that determination.