Your interior dash illumination is powered off the taillight circuit. However, the taillight circuit is different from the stop light (brake light) and turn signal / hazard light circuits.
This leads me to suspect there's a slight short circuit somewhere that results in the taillight circuit being 'cross-juiced' when either the turn signal or brake lights are activated.
Since these circuits have nothing to do with the headlights and it occurs with only the right turn signal (if I understand correctly), my first suspect would be a wiring fault (slight short) in either the right rear combination light assembly or the wiring leading to it.
The turn signal - to - taillight / dash cross-activation might also (conceivably) occur if there were a similar 'short' in the light / signal control stalk on the left side of your steering column (or maybe the switch unit attached to it). However, this wouldn't explain how the brake light circuit was also activating your taillight / dash circuit.
I'd start with wiring, etc., in the vicinity of the right rear light.
2000 model year wiring diagrams (same generation as your 1999) can be accessed at:
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or
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Hope this helps ...