I've dealt with this problem. Hypersensitive gauge is a gauge problem. If it were a simple on/off problem I'd say external wiring, but in my case, slowing down, starting to move, hitting bumps, etc., all caused the needle to bounce wildly.
All damping is in the gauge, and NO gauge when working correctly, no matter what happens to the external wiring, will fluctuate that fast. No "modern" GM fuel gauge is setup to allow fast changes in fuel sender signal to be reflected in the gauge just as fast.
There is a resistor on the back of the gauge, but as far as I can tell it was used to "tune" the gauge, since you will find them with varying amounts of resistance, and they appear to be color coded like fuses. By tune I mean fine tune the E and F position.