1987 Nissan 300ZX headlight

This model along with most Nissan models in the 90's has been known for this issue. What causes it is simply the head light switch. The switch has seperate contacts in it for left and right sides as well as high and low beams. Over time these contacts get hot and warp causing undesired contact (causing the light to come on). Since headlights are powered all the time, the lights can come on even with the engine off.

Replacing your headlight switch should resolve the problem.

I agree. 99.9% of headlight/turn signal problems are caused buy a defective headlight/turn signal switch. If you replace it yourself, $100.00 to $160.00 for the switch and more than
1 1/2 hrs. of your time.