If the dome lights stayed on all the time it would indicate that one of the 3 buttons that trigger them is defective (one on each door and one on lift gate). By removing each, one at a time and removing the wire, it should show which switch might be bad. I believe the lighting circuit is closed when a ground is created thus turning on the lights, so it could be something is grounded which shouldn't be. Do you have a simple continuity test light?
ALSO CHECK TO MAKE SURE THE NEW LIGHT SWITCH IS NOT TURNED ALL THE WAY CLOCKWISE BECAUSE THAT WILL TURN ON THE DOME LIGHTS AS WELL. TURN THE KNOB COUNTER CLOCKWISE A BIT AND SEE IF THE DOME LIGHTS MIGHT NOT BE ON