Back to the question asking if the rheostat has any impact on the lights that stay on - NO. But I did discover that the rheostat only works on the instrument lights in the dash, not these back-lit illumination bulbs. The intensity on these (around light switch, inside window switches, under gear shift lettering) is only reduced by the activation of the light switch itself. They get dimmer as soon as you turn the headlight switch to parking lights.

I have looked at the Haynes manual and found that their schematics read easier than those in the Owners Workshop Manual (published by Technibooks) that I was referring to. According to Haynes, the Illumination-dimmed as well as the Illumination-not dimmed are routed through the instrument panel from the light switch, via the light control unit. But it doesn't appear to go "through" the light control unit. So the light switch looks like the source of the problem.

Does it make sense that the switch is broken in the "on" position (circuit closed) yet the dimming action still works when the parking lights are turned on? I can't determine where the extra resistance is introduced to perform that dimming function but hopefully it is NOT inside the light control unit. That part looks like it might cost more than a new light switch.