Any time there is a difference between what you see in the Animation workspace vs. what you see in an export, it's important that you understand the difference between "video assist" (which is what you see in the Animation workspace) and "high-res" images (which is what you usually are exporting). If you are not sure about the difference, read the "Big Picture" section of the User Guide.
In this specific case, the problem is almost certainly that you have "Image Stabilization" engaged on your camera lens. Ironically, the way image stabilization engages every time the camera shoots can pull the lens into different positions, causing the shakiness in the high-res captures. Then the image stabilization relaxes during live view, and the lens settles back into a consistent position.
Note that our camera setup instructions for Canon cameras tell you to turn off Image Stabilization.