Why does my car shake?

When I'm driving if I push the gas too hard my car starts to shake. It feels like its shaking side to side if that makes any sense. The first time this happened the check engine light came on and when I got it checked it read bad O2 sensor. I got it fixed and I still can't push the gas very far down before it starts to shake. I can accelerate slowly all I want and its fine. What could that be? Another thing it does is from time to time (maybe once every 1000 miles or so) it will jerk and slow down for about 1-2 seconds when it goes from high 30mph-low 40 MPH. Its really subtle and my passengers dont usually notice it but I do and it freaks me out. I'm going to get a new fuel filter soon. Would that help?