If you have stability problems, make sure your wheel width is wide enough. Most Stock racers use 10" rims with 9" slicks. Worn rear or front suspension components can also be a factor.

I use Hoosier lightweight radial slicks. But I usually run on pretty well prepared tracks. Hoosier does have a variety of sizes to choose from and a call to them could also be helpful.

Hoosier slick specs