Check the front suspension for loose front end parts... Really bad tie rods and bearings will do that (normally on Chrysler vans)
Under load while accel, the bad toe angle puts the loose tie rod under load and keeps it from moving too much, let off the accel, then its the road turn which dictates how straight the tires are. Normally the tires will move back and forth a bit and cause the tie rod to knock around.
Check for loose stab links and blown strut mounts or loose strut to knuckle mounting bolts as well.
Axles is the next thing I would look at. When you turn pretty decently, under all the conditions you described, is the noise and vibration the same???