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Check the wiring harness, the large connector at the engine. Remove the engine cover and make sure the male and female ends are plugged in all the way.
Make sure there is slack in the boat's main wiring harness to the engine, that it isn't pulled tight.
Is the key switch in the control?
Check the fuse on the engine too. But that shouldn't affect the engine shutting off.
Post back what you find and we can go from there.
Your engine's ignition system is self contained. As long as it is running the ignition works. The ignition stops when you ground the ignition system, on your engine it is the black/yellow wire. If the engine doesn't turn off then you are not grounding the engine. Depending on which control you have your key switch and kill switch are separate but do join in the harness. The trim switch is independent from everything else with the exception of the harness itself.
So We can rule out individual switches. this leaves the harness or harness connections. The start and trim is fuse protected. The ignition cut off is separate. So the only thing that can disable everything electrical on the engine is a bad harness or connector.
Do you have any power at the starter solenoid?
Do you have power to the trim relays? Ground?
Is the battery in good condition and fully charged?