I'll make a few comments to hopefully help.
Odds are that there is a leak at the transom seal. These are often caused by the transom rotting.
Not sure about 4Winns being wood-free, but maybe. Often the stringers are synthetic, but the transom is wood core. Boat companies usually do not want you to know how their boats are made (else you might not buy them
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Another possibility is the Y-Pipe leaking. There is usually a gasket where the bottom of the Y-pipe (exhaust pipe) mates to the transom assembly. If the gasket fails, then the boat will leak at the gasket.
Another possibility, which has been touched on is a leak at the raw water intake line (hose/fittings/etc.) When the engine is running the intake line is under suction, and so it won't leak, but possibly will draw some air into the raw water flow. When the engine is off, the intake line is under some pressure from the outside water, and therefore can leak.
Leaks at/around the transom assembly can be difficult to find, as much is hidden. A borescope can help pinpoint the source. There are some scopes that are not all that expensive, but can be helpful both now and in the future.
Most often the engine does need to be removed in order to both verify the source of the leak and to fix it. The priority of fixing it depends on were the leak is coming from. It depends on what you find and can verify. A transom issue would be a high priority to fix . .