Okay, here's the latest in the saga.
In researching the gimbal, I learned that it uses MEMS gyroscopic sensors on the main circuit board. From my understanding, these are very coarse in terms of value. They also seem like they would be fragile.
I've tried to think of every possible common denominator amongst the various Phantoms, gimbals, F/C modules, remote controllers and other parts I've been using. One denominator I hadn't considered was that, with each bird/gimbal combination, I would roll, pitch and yaw the gimbal to its extremes. That's where I'd see the odd behavior.
With the idea in mind that the MEMS gyroscopic sensors might be fragile, I bought a brand new gimbal, with the intention of not taking the gimbal to its extremes. I mounted the new one, and it started up perfectly level. I started the motors and walked around the house with the bird, rolling, pitching and yawing it, but not to its extremes. It has stayed level throughout all this.
The next test will be to fly it and see what happens.