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A bias ply tire would feel like this not a drag radial. Especially if its a 20. There is simply not enough sidewall for that to be the issue, plus radials are designed to have stiff sidewalls and not wad up the way a slick does. There is most likely something else going on. You would have to run them with no air in them for it to feel the way you described.
Have you changed anything else? What kind of speeds are we talking about?
Check the circumference of the tires. Tires made on different molds and from different batches can be different sizes even though the tire size on the sidewall is the same. Mickey Thompson drag radials are racing tires and built to different standards than production car tires. I have seen many many times where two tires show up and they are different sizes causing the car to do weird things. This is where I would start.
Also, keep in mind that launching a car on sticky tires with a good amount of power is know to make eccentric bolts move. IF this is whats happening, that means your alignment has changed. It's possible the rear toe or thrust angle is out of alignment, causing the issues you're describing. Either scenario is completely plausible and possible.
Feel free to call me or or post up if you have more questions or want more info on how to check the tires and alignment.
Pete
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