Left blinker does not work when the running lights are on

I am assuming when you say running lights are not connected the blinker works you mean the lights are turned off and the plug is still plugged in.
If the blinker on the trailer works correctly with running lights off and does not work with the running lights on there is a short between the running lights and the Left turn signal. On an older trailer first place I would look is in/at the light fixture, next would be at the trailer 7 pin connecter.