Unplug one headlight [ twist connector at the light bucket] at a time and watch the ammeter. See which light, if any or both, is/are causing a problem. This is jumping the gun, but, the headlight connectors can be problematic. Thats why i'm thinking the problem could very well be a headlight connector.
Just keep disconnecting things until the problem stops, that your problem.
If the switch is turned to just the park lights, what does the ammeter show ?
When the brake lights are on, what does it show ?
I wouldn't recommend driving it with the lights on until the grounding/short issue is fixed.