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The common fault usually is the headlight switch itself. The contacts inside the switch get out of there holders and then they make contact permanently with the input contact. The only way to repair this will be to replace the switch with a new or good used switch. I hope this answer your question should you what more information please do not hesitate to ask.
No one cannot see the contacts they are inside the switch. The switch can be tested with a multi meter but it will only confirm what the dashboard light are confirming already, the power does not get switch off when the switch is in the off position.
If one unplug the switch from the wiring harness the light should go off.
This means the the dashboard light system which is also working together with the license plate lights are getting power from some where. The fault will not be easy to find now as you will have to dismantel the dash board bit by bit unplugging the wiring from the switches and instrument cluster. one by one to find out where the fault is. Very hard for me to try and describe on how to do this. The fault might be unit which works on the dash board light system.