I once had that problem on my 1987 4.3 Cobra. It was aluminum corrosion of the y-pipe at the gasket seal. The reason that the leak stopped when the engine is running is that while running, exhaust pressure kept the water out.
If the drive shaft bellows is leaking, then it will still leak with the engine running...although it might get slung about so you are not seeing it.
I had to pull the engine and replace the y-pipe.
Take a very close look with your phone camera or mirror at the top of the y-pipe to see if the water is coming from the gasket seam where the y-pipe bolts to the transom bracket.
Once the Y-Pipe is removed, it might be repairable by welding and machining. Or it can be replaced, but it is likely that the OMC y-pipe is not available. I had used a new Volvo Y-pipe, but I had to modify it slightly by denting in the pipe where the shift cable sheath came in contact with it. (the OMC pipe has a notch there. the Volvo does not). I also had to make an adapter plate because the shape of the gasket surface is different (although the hole pattern is the same). The adapter plate should be made from Aluminum, at least 1/8" thick. I made the mistake of making a SST plate the first time I did this repair. The SST plate caused galvanic corrosion of the aluminum surface, so after a few years it started leaking again. So I had to pull the engine again and machine the gasket surface and make a new aluminum adapter plate.
Ray
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"Knot Easy" 2000 Horizon 240 Volvo 5.7GS /SX
tow: 2017 Honda PILOT EXL-AWD
prev. boats:
'87 Chaparral 198CXL 4.3 OMC Cobra
'69 Jetstar 16ft Ski Boat, 115hp Yamaha
'68 Aluminum Jon Boat, 3hp Sears
'64 Water Wings