Often when I move an artboard in Illustrator using the mouse, the contents end up having moved half or 1 pixel within the artboard. Is there a setting or something else I can do to prevent the artboard's contents from moving in
Make certain the "Move Artwork" option is not depressed before moving the artboard.
It will be a dark button or a light button. You want it to be a light button if you want artwork to stay in place while the artboard moves around under it.
Edited to add:
In addition, you'll want to select the artwork and check the Align to Pixel Grid option on the Transform panel.
Art may shift and snap to the nearest pixel if this option is selected. It's designed to help prevent anti-aliased, blurry, strokes when creating content designed for the web.