Less than 12v is a pretty discharged battery. I would expect to see 12.6 to 12.8 volts at the coils unless the headlight is on at the same time. Can you jumper it from a fully charged battery and see if the problem goes away?
Also, originally your post mentioned the white wire from the coils to the kill switch. Did you figure it out? I'm pretty sure it remains attached to the coil.
Cheers,
Shawn