The power supply is a wall wart rated 12V/1200mA.
I haven't measured the voltage with just the lamps on, but can do that tonight and report back. I couldn't find 12V bulbs locally, so I just went with the 6.3V bulbs. Pardon my ignorance, but do the 6.3V bulbs pull twice as much current because of the 12V rail?
Should I install series resistors for each of the bulbs? Would that help? If so, what value?
As you can see, I know just enough to be dangerous. Thanks for the help so far.