Have you checked your tire pressures? If they are too low the tire can lag behind the steering input and make the car kind of lurch as the sidewalls flex and then stabilize to turn the car. Newer tires are typically designed to run higher pressures than the vehicle manufacturer recommends. The tire guys are after performance and life, while the vehicle manufacturers are after that "Boulevard Ride".
The sway bar would take some of the "slack" out of the suspension and should cause things to react quicker. Generally they should help, not hurt. On their own, a bigger front sway bar will add a degree of understeer to a car, while a bigger rear sway bar will add a degree of oversteer. That is why most cars only have a puny rear sway bar if they have one at all. Understeer is MUCH safer than oversteer.
How many miles on your ride? It might be reasonable to give the suspension a good looking over to make sure something isn't worn or loose.
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