I am using a precision Bourne 5K pot and I believe no slack in the gearing.
It is obvious that the hunting comes from the cheap servo. The servo acted this way without being connected to any gearing anyway. So before I shovel out a bunch of money as high torque servos can be pricey I am wondering if this is just a cheap chinese servo problem.
Sorry, you didn't / don't mention the specific servo you hacked and the comment in your OP about cheap servo was followed by mention of two fairly respectable servos. If you are using, as zoomkat suggests, an MG995 or any other variant of the same manufacturer of that servo then you've found your problem. Alan's comments about a/d resolution directly apply if you are using a digital servo, however even if you are using an analog servo controller the increased angle range will make even a decent quality analog servo more susceptible to jitter in the pwm control signal (which could result in oscillation).
Also, digitally speaking, even a 13-bit a/d converter will resolve to more than a degree per bit. It is unlikely any off the shelf servo has a 14-bit or more a/d converter so unless you are able to wire up an analog circuit you might be better served to use a small microcontroller with a built in a/d and wire up a small h-bridge to run the motor. There are several modular products avalable that can help reduce the amount of "assembly" required if you search about.