Thank you, everyone. As Charles Barkley would say, I might be wrong, but I doubt it...
Four headlight systems have a single filament (either a high
or a low beam) in each bulb, while two headlight systems have two filaments (both high
and low beams) in each bulb. None of the Eagle's headlights would have two filaments. When replacing these bulbs, you'll need either a high-beam or a low-beam. Again, I might be wrong, but in all of my years of working on cars (more than I care to admit) I've never heard of a dual-filament bulb in a four-headlight system. If the outer bulbs had both high and low beam filaments, that means you'd have four high beams plus the low beams going when the highs are switched on; I don't think so...
I suspect something is loose or broken in the high beam switch, but since we're planning on replacing all of the headlight bulbs anyway, we'll be able to clean the sockets and rule these out. Between my job and Joe's classwork, the weekends are pretty much the only time we'll have to work on it, so we'll report back after this weekend.
Thanks again.