My first hand experience with a stuffing box which was so tightened up that it did not leak meant it was not properly water lubricated and it was grinding a groove in the surface of the shaft. Unless it has been filled with the teflon impregnated packing or one of the silicone putties which form closely round the shaft and don't leak yours may be doing this too.
When the engine is driving the boat the engine and propshaft moves forward a little on the AV mounts so the wear from the packing is slightly nearer towards the stern. Thus, when stationary, it resumes its previous position on an unworn part of the shaft and this fore and aft movement over differing diameters gives rise to an ever increasing rate of leakage and you will eventually need a new shaft - as I did.
You could try unscrewing the compression nut and picking at the packing to identify it but you would probably make it leak! The fact that it runs cold is a good sign but you need to run in gear at cruising revs for 15 to 30 minutes to be sure it is not heating up.
Can you enquire about the packing from your previous owner?