Most likely the headlight relay/wiring harness has melted.

The relay is in the dash under the two central vents with the power lock and power window relays. Easiest access is to take the center console off. There are 2 screws under the coin tray, 2 on the top side, 2 at the bottom under the storage box, and 2 clips on the bottom side (use a dime to turn and release). If you pull the cover back and look up under the vents you should see 4 metal relays. The one closest to the drivers side is usually the headlight relay. You should be able to identify it because the harness will probably be melted.

You may need to repair the wiring harness, and replace or clean the relay. I just cleaned the one on my 93. You can take the cover off and clean the carbon off the relay contact.