I remember asking about lights on my 90 Bonneville a while back and I was told there are no relays in the headlights. Not sure about the Buick. If it is the same, the problem sounds like a bad dimmer switch, bad wires or corrosion on the bulb hook ups. If I were you I would start by checking/cleaning the bulb plug ins, verify power at that point, and work back from there.
Is there any way possible both bulbs burned out at the same time? I used to work in a liquor store and people would come through the drive through that had no idea they had a headlight burned out. So then I sold them alcohol.
But really sometimes it is hard to tell if you are driving in a well lit city all the time.
On the old sealed beam, you could hit it with the side of your hand a couple of times and the filament would contact and start to burn again. It is usually enough to get you home. I like to believe I saved a couple of tickets that way. But these "new fangled" bulbs don't seem to respond to the Fonzie touch.
I'm sure everybody else is tired of you high beaming them too.