Hi Jessica,
Mosaic datasets use overviews when you view them. Be sure you copy the overviews to the new server as well. The overviews are found in another folder in the same directory as the geodatabase with the following name:
<geodatabase name>.Overviews
When you zoom into large scales, overviews are no longer used. The raw imagery is read from disk. So, when loading your rasters to a mosaic dataset, be sure you are using UNC paths (i.e. \\<server name>\<folder>). That way the new server will know where to find the rasters when you zoom into a large scale.