Sounds possibly like sticking contacts points in the main relay, or possibly a faulty diode in the relay unit allowing current to flow backwards?

Check is the main relay, which sits next to the starter relay. It's the biggest of the two relay boxes you can see under the seat behind the battery. Take it off and have a look at the connector for corrosion or any other problems. Also you could try drying it out inside for a while?

You should hear it click when the ignition key is turned on and again when the key is moved to the off position.

It could also be a short in the ignition switch itself, or in it's connector?

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