Steve's probably right. This requires getting a good look at the schematic for a '68 GT Mustang.
Both the foglights and headlights will also turn on the taillights. It's somewhere in the middle (a la short) that's got that one foglight to come on.
Get a good look at how the foglight wiring is. The pigtails to the foglights (same as GT lights) are part of the same harness as the headlights. A wire inside that harness might be touching "something".
If this continues, you might fry the innards of the headlight switch.
Keep us posted as to what is happening. Headlights and fog lights are items that work without the key turned in the ignition.
Paul N.