'66 Sprint GT, '67 Duetto, '70 BMW 2800CS
I have converted two of my older cars to LED bulbs and electronic flashers and neither exhibited the symptoms you describe.They all work rather well without the engine running. However, after starting the engine it causes the indicators to go erratic. Flashing at and increased rate and a little bit erratically.
The operating voltage with the engine off is lower (~12v) than the voltage with the engine running (~14v). So something about that higher voltage seems to be causing your electronic flasher to go nuts.
Are you confident that the flasher is wired correctly? Is it possible that your flasher unit is just defective?