Ignore the alternator pigtail, it's just fine.
Fix the headlight pigtail first.
The way it sounds the headlight is grounding out. The ground voltage is actually back feeding through the headlight switch, using it's ground, which is being overwhelmed. If left too long a fair many things will suffer severely. So fix this fast and with driving the car as little as possible, and not using the headlights.
Don't "think outside the box".
There is no box.
Do it correct, or do it twice.