Electrical fault

My trusty old '95 850T has just developed an electrical fault which, I'm afraid, will probably turn out to be expensive to diagnose and fix.
When I turn on the headlights, the instrument panel lights up for about one second then goes dark. If I turn the lights off and back on again, the panel doesn't light up at all. However, if I switch the engine off and re-start, and then turn the lights on, I will usually (but not always) get the one-second treatment.
Any insights -- or even hunches? I'm not very electrical. Could this be a failing capacitator or something? Any notorious examples of this sort of thing in the lore?
Grateful for any comments or suggestions.
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