Hey N15MNO, as the others have said, it's a good idea to learn the full capabilities of what your 350 will do. Have you driven a few other 350's as a comparison. I'm not a fan of the stock tyres that come on our cars, but it's a personal thing, so I'm on my 2nd set of aggressive directionals now. Tyres obviously have a significant bearing on how a car handles.
I've had a hell of a lot of handling problems & other mechanical problems with my 350 (it's a very special car ). I've dropped out the stock suspension, and replaced with the Tein setup. I'm very happy with it thus far. It's improved the handling, stability, and ride quality no end and is quite low key hidden away.
Don't forget to let us know how you are getting on.