Quote
Voltage at battery (when engine off) 12.35v and 11.30 at the tail light socket (parking light pin)


12.35 is kinda low, I predict a new battery will be needed next winter.

11.30 is a lot of voltage drop. I agree with the folks saying check *all* grounds, also unplug & check any connectors you can get to.

Pay extra attention to the circuits feeding the tail light relay, if it's buzzing, it isn't getting enough current to pull the switch fully.

Dave