The swaying is typical of bias ply slicks, regardless of the type of tire is on the front. The only time mine has swayed bad on bias plies was when the pressure was too low.
On a heavy car, you'll probably be around 14-15 psi for them to be happy.

So your car only picks up 15 mph on the back half N/A? And then only gained 6 mph at the 1/8 on a 350 hit???? You have some issues going on here.