Charging system issue

[marauder1313]

So I just got back from an 18,000km trip with this GZ, and something that I experienced back in Delaware and had to live with all the way back to vancouver was an issue with its charging.
With the headlight plugged in, im barely getting 12.6V at idle. With headlight unplugged, Ill get into the mid 13's

I actually ended up riding back to the west coast with my headlight unplugged because I would encounter a dead battery eventually.

I got a brand new battery at an autozone in philly too

Where should I start looking to figure this shit out?

Thanks