2002 Chevy Cavalier Car Violently Jolts When Shifting Gears

I was driving my 2002 Chevy Cavalier LS Sport Coupe powered by a 2.2 Liter Ecotec Inline 4 Cylinder. I drove into town to go to Wal Mart and the car was doing fine. Once I left, I noticed that when I moved the shift knob into "D" (Drive), the car made a slight jerk. It wasn't that bad at that point. However, when I started driving, I noticed that when it started shifting through the gears, the car would jerk every shift. An amber light that says "Trac Off" which means tracion control off, came on which is not normal when the car is in drive. I made it home thinking it was just a plug that fell out or something. However, when I started the car later that night, it was the same thing, but this time the jerking became more and more violent to the point that I was able to see the engine actually jump everytime I moved the knob into drive. I decided to let the car idle for a bit. About 5 minutes into letting the car idle, the check engine light came on. I am confused about this. Could it be a transmission problem or is it simply the motor mounts going bad? I am clueless here.