I know I'm late on this by a couple months....but the problem sounds like front end toe-out, (front wheels both turned outward a couple degrees) which makes the front end dart all over the place and try to take a steering line of its own. The worse it is the harder it is for the steering wheel to return to center.
Also a tie-rod end. If the steering tries to keep pulling itself around that might be from one wheel being loose. Check that by jacking and grabbing the tire at 9 and 3 o'clock and trying to wiggle.
Finally,. wheel bearings can be deceptive. If they are worn they can cause the wheel to wonder around. Check by jacking and grabbing at 12 and 6 o'clock.
The geometry is pretty precise and bearings, tie rod ends and alignments are critical. The ONLY type of alignment for a vette is the 4-wheel-thrust angle alignment. This takes the center line of the chassis and sets ALL 4 wheels to track in the same direction, same line,. 0 degrees off the chassis line or whatever the factory spec is. On a vette, its not much. With tires this wide the castor and camber have very tight specs.
And the strange part.....The rear end is just as complex as the front and can cause similar issues. A loose rear toe-rod end castle nut used to make a c-4 that I had, track funny during accelleration, then snap back in line when you let off the gas.
