67 barracuda rear lights stay on

In addition to above, could be a few things. Rust, corrosion, malfeasance under there in the pedal hangers, the master, etc.

Bad/ misadjusted switch

BUT BUT DAMMMIT BUT................the switch hanging up should not cause any wiring to get hot. CHECK the bulkhead connector for poor connections. Something is generating that heat

I just HATE malfeasance..................