If I remember from my teenage years: If you're looking for traction, you need wide tires with good grip and stiffer suspension. Way back I added sway bars in the back, that car (rear drive Ford) drove like on the rails, I don't think larger diameter tires would do that much to the soft ride Toyota has. In hard cornering TCH pushes like crazy, I could not break the rear tires, even on ice without engaging parking break, so maybe that's the sloppy feel you got, when you're loosing traction in the corner the front will slide and keep car going straight, rear will follow and will not slide out even if you tried.
BTW If I remember correctly Camry max out during cornering at .78G, that's lower than Accord .8G and much less than Nissan I used to have with at least .9G. As a matter of fact first week I switched from Nissan to Accord I almost wiped out on the entrance ramp, when rear got loose, with no warning.