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I have a 1981 f700 dump with this odd problem:
All lights work fine with engine not running (also fine with key on)
With engine running and running lights on the right turn signal has dim flash every 15 seconds, left signal and all other light work fine with engine running.
Same problem with the trailer hooked up.
sounds like you have a bad ground or a bad bulb up front. there are turn signal grouns right by the marker lights, check those because they back up / get stripped pretty easy.
What about the hazard flasher? If you select the hazard flasher switch, does the right ignal work fine?
The problem might be the Turn Signal Flasher.
Usually there is a separate flasher for the Hazard and a separate flasher for Turn Signal.
Slow or no flash on one side is usually a sign of a bad bulb on that side, but I recently found a bad flasher causing the same problem on mine. One side would occasionally flash slow or not at all. Apparently the flasher was "borderline" and the slight difference in resistance between the two sides was enough to cause the problem to only show up in one direction. Give it a try. They're cheap!