To RADIAL or NOT to RADIAL?

[69CudaFB318VA]

I have a 69 Barracuda Fastback 318 w/automatic.  I put radial tires on it but the back end feels very mushy when turning a corner.  It feels like the back end is swaying to the point that it feels like oversteer. 

Radial tires are much better than bias ply but with the poor handling, I wonder if I need to change shocks or other suspension setup, or go back to bias ply.

I'm looking to largely have the original look and feel so which way should I go?  Thanks in advance for the advice!

[Racer57]

I have BF Goodrich Radial T/A's which is what a lot of people are using. Right or wrong, I keep mine at 40lbs which does make my 70 Cuda wallow around a lot less than they would at the 32 +/- that their suppose to be at. I really like how mine feels with them. Interstate off ramps that are posted at 30mph, I've taken at 50 real easy. 

[69CudaFB318VA]

Thanks for the reply.  I may give that a try.