Headlight problem

Hese cars are know for getting water in the headlights. This causes electrical problems, and at the very least blows bulbs. When you turn on the high beams, the low beam turns off. You may just have a bad high beam filiment. Try replacing the bulb, and check it for water. If it has water in it, it should be replaced. I have seen people try to seal them back up once they are dried out, and I've also seen drill small holes in the bottom to allow the water to drain out. But, replacing them is the best, however most expensive option.

Do both headlights turn off or just one? Did replacing the bulb do anything? You made sure not to touch the bulb with your fingers, right? If you touch it, it will blow intantly. If you're good on all of that, it may be the headlight relay.

Assuming the bulb is good, check the plug that plugs into the bulb. Get a volt meter, with the lights on, make sure you find the normal light wire and see that you have 12V, as you know this one is good. Then switch it to high and check the other wire for voltage. If none there, you'll need to trace that wire back to any plugs until you get to the solenoid in order to find where you are losing the power. Could have been a mouse snacking on a wire, or just a loose or corroded connection.

It wouldn't surprise me that the plug is melted if it did have a water problem in the headlight.

I think it could be any connection or wire if the bulbs you put in drew too much amperage.