The first step is to determine whether it is the display shaking, the image shaking, or the video information being fed to the TV shaking. Checking the menu to see if it shakes would eliminate the display if it does not. If the menu still shakes try unplugging everything from the TV. This would leave just the TV connected to the AC. If the menu still shakes then check to see if there is anything else on in the home that is connected to the AC that can cause noise on the line. You want to eliminate all possibilities so you can be sure you have a problem with the TV.
If the menu was fine when it is just the TV connected, then something connected to it could be causing the video to shake. Connect the cable box and the XBOX one at a time by themselves to see if you can determine which one is the culprit. If, for example, it only happened with the cable box, try another output. The cable box should have RF, composite, component and HDMI to choose from. If it is on only one type of connection, try a different cable. If they all do it, then the video board would be suspect because it is the common factor between all of them and the menu is added after most of the video processing. This model does not separate the video processing form the A/V switching so it would be all on one board.
If you eliminate your home's electrical system and all devices plugged into your TV but the screen still shakes then we have to point the finger at something inside the set. The first check I would do would be to look for vented capacitors on the power supply. A capacitor has a round flat top and when damaged will start to bulged up a little. A technician can spot these and replace the caps with the same value at a little higher tolerance to prevent it from happening again. The other check would be to remove the LVDS cable that sends the video signal from the main board to the LCD control unit. The contacts on the cabling could be contaminated or making a poor connection. These checks are not meant to be performed at a consumer level. You should find a repair shop or call for in home service if you are unfamiliar with the inside of a TV.
If for any reason you need to order a board for this TV make sure you find the correct match for your serial number because the parts are different.