If you car is shaking get your tires balanced. Balancing will not have any effect on pulling.
You shouldn't need to get an alignment when you rotate your tires.
You would think they would make sure the tire pressure is correct when they rotated the tires but check it anyway.
You could try swapping the two front tires. That may take care of it.
I just had mine rotated and thought I noticed a little drift that it did not have before. A few days later I don't notice it any longer.