I didn't catch the 3V setup xD
That could have been even worse!
I do think you should disconnect it from the car though, gut the new light fixture and re-do it from scratch.
You can use a pair of 6V or 12V incandescent lights and simply wire them to power. (In series or parallel respectively)
Or you can take the LED route... either buy an automotive LED 'bulb' for a dome-light, or build your own from scratch.
If you build your own, you will end up saving a bit of money... the LEDs and resistors will most likely cost you less than $3 .. The trick is putting the right value resistor in-series with your LED(s)
If you do the DIY method... go buy two white LEDs, but be mindful of the current draw.
== Math Ahead! ==
First fine the Ω value of the resistor you need.
[ SupplyVoltage - sum(LED1_Vf + LED2_Vf + ...) ] / min(LED1_Current, LED2_Current, ...) = ResisterΩValue
Then find the power handling you may need.
[ SupplyVoltage - sum(LED1_Vf + LED2_Vf + ...) ] * min(LED1_Current, LED2_Current, ...) = ResistorPowerValue
If you use two white LEDs rated for 3.3V @ 20mA...
Then you would need 370Ω resistor rated for more than 148mW ... of course, you can't buy that one exactly... so pick the nearest larger value for each requirement.
Connect them all in series, and connect to the wires of the car and you are good to go ^^
*Please note: Although your car is a '12V system', you actually end up dealing with a noisy 14.4V supply voltage when the motor is running. (Alternator provides 14.4V to charge the battery)
This may also be quite noisy, so if you plan to use sensitive electronics, you should concern yourself with filtering.