splice them into your existing dome light switch rather than the lights under the dash(?)...the dome light comes on when you open your doors, correct? It goes out when you shut your doors, correct? But you can also turn on the dome w/ the switch when the doors are shut, correct? Remove the actual dome lightbulb if you want to, but make a complete circuit w/ the wires using the neons as the loop rather than the dome light or find a cool colored bulb that compliments your neons.
Or go buy a junker dome light relay & doorjamb switch from a junkyard & make another circuit...