I've got those Avons on my bike and simply changing the tires and truing the rims made a night and day difference in the handling. The previous rear tire was almost perfectly square and the front tire was a Cheng-Shin, so obviously putting any new tires on would improve the handling. I have a lot more confidence on these tires and the modern tread design helps to keep them from weaving and wandering on grooved surfaces.
I'm currently updating my suspension, so the handling should get even better, but the tires themselves made a world of difference.