There's MOVING and there's COPYING. Moving does just that - it takes the files from one location and puts them in another. Copying makes a COPY of the file for the new location. In other words, if you MOVE the files you'll only have one copy. If you COPY them, you'll have two copies. And in that case, MB will NOT remove anything.
If you select the files you want to MOVE then right-click > send to > you'll see folder options for moving or copying. Both of which can involve organizing or not. If you're not sure what organizing does, read the Wiki before you do anything. And when you are ready, start with only one or two albums as a test to make sure it's doing what you want.
If you move/copy the files outside of MB, say with a file explorer, you'll have to rescan everything as MB won't know where the files went.