Did you replace the transmission range sensor? Is that what you meant when you said neutral safety switch? If so, here is what I suspect is happening:
The left rear reverse light shares a ground with the indicator on the shifter. I have a feeling either that ground or the ground for the right side is bad or weak. When the bulbs are in and require the ground, you lose the light inside and opposite.
I attached the wiring schematic below and highlighted the circuit and grounds. I need you to locate and check the grounds. To me, the one of most interest is the one under the dash shared with the left reverse light. However, the other could be faulty as well.
Here is a link you may find helpful:
Let me know what you find or if you have other questions.
Note: pics 1 and 2 are the schematic. I cut it in half to make it readable. Pics 3 and 4 show the ground locations.
Images (Click to enlarge)
Saturday, July 30th, 2022 AT 11:26 PM