It may be that the red hazard switch spring or contact at the relay is worn out and allowing contact to the relay in hazard and turn operation, while the contacts to the lights were still able to disengage in hazards off.
The relay is solid state so it is transistor switching, not a coil. Your relay seems to be getting power and ground and the red hazard button switches between fuse 1 hot always, and fuse 16 hot in run or start. See if the relay continues to click with the key off and it's probably at the hazard switch.
This little switch controls the function of practically every exterior light and the contacts are really close together.
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I've never tried but apparently the switch can be removed without taking off the entire turn signal assembly.
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