Also what is surrounding the shaft log? Meaning in the part of the boat you can't see from the prop to where it enters the cabin near the stuffing box? Is it hollow? is it fiberglass? I'm trying to figure out that if water is getting outside the shaft log, what it might be damaging
That shaft log could be going through solid fiberglass, or the builder may have filled the void with some other material such as ballast, foam, or something else. Without knowing what the builder did, I can only make an educated guess, and I would have to agree with another poster that they probably used a fiberglass tube as the shaft log, and probably not through solid glass (too expensive). If that is indeed the case, then inboard end of that tube has to meet the fabric layers of the hull where the stuffing box is mounted. That's a somewhat awkward joint to make with fiberglass.
Since the leak only occurs when you have the boat under way, clearly the flexing of the hull from the force of the propeller pushing on it opens up the leak path, and cutting power pinches it off. It's far enough below the water line that hydrostatic pressure will cause the leak to spray, as you have observed.
Putting it together, here's a guess: There's a crack at the inboard end of the fiberglass tube/shaft log that opens up when the hull flexes under power.
If it were my boat, I'd keep an eye on it through out this season, and then address it after haul-out in the fall.