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Welcome, I'm Chris (aka Moose).
While running what is the battery voltage?
Do the headlights work with the engine off?
At the headlights are blue and black wires. Can you ohm test the black wires to ground to verify they are good and can you see if the blue wires have power when commanding low and high beams on?
As long as the battery voltage while running is 13-15 volts, then the issues do not appear to be alternator related, especially if you have issues with the key on, engine off.
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