You should be able to save after you preview. If that's not the case, can you give us a screenshot?
Regarding the file staying in place, reorganizing files is a completely separate operation. Infer & update only changes the tags. In any case, it sounds like your files are properly organized by album and artist, else MusicBee wouldn't be able to find those values. So even if you have auto-organize enabled, it may not need to move any files.
If you want to add artist and album to the file name, you'd need to change your organization template. Don't do that while you've got missing tags, though, poor you could lose the info of the file structure.