I suspect it's the headset installation. The type of squirrelly steering you describe is characteristic of a tight or binding headset. That makes it stick/slip as you make the tiny corrections necessary to balance a bike. The handling along with your headaches adjusting point to a headset issue.
My first guess is that one or both bearings are upside down. Or a spacer is missing or something like that. Pull the handlebar stem, drop the fork and take a look.
One other possibility is the steerer rubbing in the lower bearing cup. This used to never happen, but these days some forks have the crown seat bulge extend too high, and overlap in the sleeve area of the lower cup. A high bulge combined with a thick walled headset will bind like yours. Test for it, by removing the fork and seeing if the bulge extends above the crown. Also insert the fork into the frame without the lower bearing and see if it's tight or even rubs the least bit.
If you have a fork/headset rub, the remedy is to use a reamer or half round file to open up the headset's ID enough to resolve it. Note: YOU CANNOT SAFELY REDUCE THE DIAMETER OF THE STEERER CROWN BULGE. SO DON'T EVEN CONSIDER IT AN OPTION.
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FB
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Last edited by FBinNY; 09-08-13 at 03:35 PM.