Definitely the potentiometer in the Servo has a dead spot. What is happening is the servo cannot Sense where neutral is exactly do it keeps going until it see's it's gone too Far then Has to Track back and then over shoots again.

The only DIY solution would be to desolder the pot and replace it. but that is if you can get the exact parts (or have spare servos with other problems)

Or just swap it out and talk to the Manufacturer about replacement.