Dumb question, are you 100% sure it's not the headset? If you temporarily overtighten it a bit, and go test the brakes, do they still jump? Obviously you can reset the headset to a proper adjustment afterwards. If it were me, I'd just like to know beyond any doubt that it wasn't an adjustment issue I was making.

Is the brake caliper attached tightly to the frame?

Are the brake arms on the caliper tightly secured? Sometimes the arms can be loose on the center bolt and cause the symptoms you're experiencing. There are different brake designs, but there should be a little locknut thing on the front, no?