Hi G. Thompson,
The reason a lamp stays on when the control switch is not turned on is because it is getting the voltage it needs to turn on past the switch. In other words, it is a short to voltage. However, the source of this voltage is only hot with the key on and not hot at all times which is why they only come on when the key is on.
I attached a few links that will help you diagnose this. The diagram I attached is of the rear marker lamps which is what I think you are referring to when you say rear running lights. However, in the event you mean the rear tail lamp assemblies, I attached that diagram as well.
The idea that you need to follow is try to narrow down where the short is located. So you want to start testing your voltage with the switch off and the truck on so that the lights on are when they should not be. Test your voltage at the lamps and then start moving forward in the circuit, measuring at easy to access points like inline connectors or fuses, until you find that you do not have voltage any more. Once this happens then you past your issue. At this point, you can start zeroing in on it or just overlay a new wire if the only thing you past was wiring. Just make sure when you overlay a wire that you cut out the old wire but cutting it and splicing in a new wire. Then you can just secure the new wire with zip ties in a safe location so that it doesn't get damaged.
https://www.2carpros.com/articles/how-to-use-a-test-light-circuit-tester
https://www.2carpros.com/articles/how-to-use-a-voltmeter
https://www.2carpros.com/articles/how-to-check-wiring
Please let me know if you need more assistance. Thanks
Kenny
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Tuesday, April 9th, 2019 AT 9:31 PM