A few Internet searches later, we figured out the problem: my long hair confuses the phone's proximity sensor, causing the screen to switch on even if the phone is still to my ear. If you're unfamiliar with the term, a proximity sensor is the part of your smartphone that detects when you're holding the phone to your ear, shutting the screen off. Once you move the phone — presumably away from your ear — the proximity sensor tells the screen to switch on. But with the Droid, if your hair gets in the way, the proximity sensor detects movement, switching on the screen and leaving it unlocked for accidental button pushes or hangups. The iPhone has a proximity sensor, but I've never experienced this problem.
It's easily solved; just be sure to pull your long hair away from your face and ear before making a phone call. After figuring out the problem, the accidental hangups stopped happening.