car stutters when giving gas

Sounds like 2 problems. When was the last time any ignition parts were replaced(spark plugs, wires cap, rotor, ignition module, etc.)? If has been more than 50K miles, have the car given a standard tune up. That should eliminate your stutter. As for the shimmy, the car probably needs some front end work and maybe tires. Alignment, balancing, and suspension members are all suspect. Have a reputable mechanic go over the front suspension and see what he finds.