Thanks for the tips guys!
So, since I have to lock the door with the key, I'll have to turn on the ignition within 10 seconds of opening the door? Every time? That is going to get old! Good thing the ignition is on the left of the steering column.

I'm almost afraid to ask, but how much is a new key that can lock/unlock the doors remotely? Can the alarm be disabled without causing other problems? At least until I can get a new key.
I did some reading on the headlights, and I think I may have a bad switch. I replaced the bulbs, checked the fuses, and pulled the switch and sprayed electrical contact cleaner into it. I also put dielectric grease on the connector. No joy. I can pull the knob and both front and rear fog lights work, and if I pull the turn signal lever toward me (flash to pass), I have high beams. But if I push the lever forward, I have nothing. I haven't noticed any other electrical weirdness, which may point to a bad ignition switch. Which leaves the headlight switch as the likely culprit. The best price I could find on one was about $100. I'd hate to buy one and have that not be the problem. Is there anything else that it could be?