I am thinking you may have a ground issue somewhere in the lighting circuit as things appear fine until you turn the lights on. The lights may be grounding through the instrument cluster thereby causing your problem.

You can try to identify which circuit is at fault by disconnecting the body harness at the Kick panel area or forward lamp harness along the fender well. It may also be the instrument lights themselves not getting a good ground, Pull the fuse.
Try the above 3 things and see if one makes the problem go away. You will then know if it is the lighting circuit and if so which one let us know