You could have a leak in the shift cable bellow or the Y pipe could have
rotted out. I don't know why running the engine would have any effect since
both items are below the water line. With the engine off, tilt the drive up
and steer side to side. Notice if there is any change of the leak rate. It
would be helpful if you had an assistant for this test.
Jim

Thanks. Doesn't look like the shift cable boot. I'm starting to think
it's a rotted Y pipe. Stuck a hose in the port exhaust hole of the
stern drive and water leaked back into the boat. I'll try your
suggestion.