the connector X1739, the one that goes into the handle and the license plate lights, has 4 pins - one common ground, one for the handle and one for each light individually. the lights do not cross into each other, each one is individually powered... before scrapping the handle module, make sure that the module is defective:
if you have an ohmmeter (or multimeter) test the continuity (resistance) from each pin to the lights - the pin out is as following:

1 – left license plate light
2 – trunk lid release button
3 – ground (brown wire)
4 – right license plate light (also feeds trailer module for parking lights for trailer)

if the continuity is ok and there is no cross circuit shorts, then your module is ok... the power to the lights is not provided unless the LCM detects functioning light bulb - during engine start up (ignition hot) the LCM sends out PMW (pulse modulated wave) through out the entire car to determine the continuity of the light bulbs. The light bulbs that are deems defective (burnt out, missing...) do NOT get voltage from the LCM until the next power cycle. put is a fresh light bulb and try the engine again... hopefully, it will work... if not, then it is time to troubleshoot further...