The shocks only go up and down and have no side to side support function at all. I think your trying to feel the difference with the new shocks and are noticing things that have nothing to do with the swap. The side to side is most likely alignment related. Side to side weaving will feel sloppy if the front bushings are shot and will contribute to some wandering of the nose under heavy braking. If something is off in the rear of the car it usually becomes obvious under panic stops as in the tail of the car will be unstable and move from side to side. This assumes it's dry outside, no snow or water dude! Wearing the inside of the tires is a camber issue. Sounds to me the first step is a trip to an alignment shop.

Les