Just a theory:
Maybe the trem is pitching forward and moving as you fret with your hand, and not the neck, and that's what's causing it to go out of tune.
After all, it's a free floating device that uses simple springs to keep it in place, and therefore in tune. Metal springs will inevitably lose some of their ability to resist stretching over a period of many years. As you move your hand on the fretboard, this puts pressure on the springs as the strings move toward the frets, and if they have lost some of their tensile strength, you could be unknowingly pulling the trem out of its normal tuned position.
If your guitar has tremolo springs that came from the factory, I could imagine the springs needing some adjustment, or even replacement, after 22 years.