I did a little bit more testing, and one thing I tried was to switch the connectors around on the 4-in-1. As it turns out, any servo that is plugged into channel 6 exhibits this strange behavior, and any servo plugged into the other channels works fine. So it's not the servo, but seems to be the 4-in-1.

Could this be caused by a bad gyro gain or tail gain setting on the 4-in-1? If not, any other ideas?

I should mention that this heli has never been flown. It was powered up once, and my wife could not get it to lift off the ground. I finally got some time to look at it, and this is the first thing I ran into.