Greetings Turbogto,
The car should be stable with the tires hooking. With the G-body, there will always be some directional instability as the car goes over bumps, but nothing that should even remotely hint of danger of losing control, as long as the car is being driven nominally in a straight line.
The problems you mention indicate almost certainly, a significant departure from the car's stock, normal behavior, even accounting for the car's 12-sec potential. You therefore need to determine all ways in which the car's current condition differs from new, and at the very top of this list, as you've already started to find, are all the bushings: both body/frame, and suspension. The body/frame bushings are notorious for the bolts seizing and thus snapping after a decade or more of exposure to the elements. (Btw, yes, you'll have to cut the trunk floor to access the top of that bolt anchor. After that, you can make a metal lid that you can screw into the existing trunk floor surrounding the hole to hide the evidence. Once you replace the floor carpet, no one will be any the wiser.)
While many would recommend using bushings that are stiffer than stock, I'd argue that replacing them with NOS bushings will eliminate somewhere between 99% and 100% of the trouble you're experiencing. Incidentally, when the bushing bolts snap off, the frame itself is usually significantly rusted, and may therefore be compromised past the point of serviceable integrity. Inspect carefully before throwing good money after bad!
HTH,
MAP