While you can probably keep your library organized with MusicBee, at least initially, this is what I'd do:
(I am assuming that you are using a Windows OS, but it should work for Mac or Linux too.)
1) Set up a new sub-directory (I use My Music and it is in the root directory e.g. C:\ or in my case, where I have multiple hard drives, G:\). Then under that directory, each "album" is a sub-directory of My Music. If you have multiple albums by one artist or band, you might want to first have a sub-directory for that artist, e.g. U2 or Rolling Stones, and then place all of their albums as sub-directories under that artist. For individual song that aren't part of a larger group of songs belonging to an album, you can always either leave those in the root directory of My Music, or a directory called Misc or Miscellaneous Songs, or group them by genre, eg rock; or grunge or whatever.
2) Then search all of your drives for your music, and it's accompanying art files. If they are already in sub-directories by album, just cut and paste them from their old location to your new My Music directory. If they aren't at least temporarily, cut and paste the undirectoried songs and art to a directory called "Need to Organise" so that you can later sort them into a useable fashion.
DON'T FORGET TO RESCAN YOUR COMPUTER, AFTER MOVING ALL OF THESE FILES, AND CHANGE THE SEARCH CRITERIA TO JUST MY MUSIC,
AND CLICK YES TO DELETE FROM LIBRARY THOSE FILES NOT FOUND AT EXPECTED LOCATION AT THE END OF THE SCAN.
3) You said that many of your files were duplicated, triplicated etc. I've been using a neat little FREE utility Heatsoft Clone Cleaner Litehttp://download.cnet.com/Clone-Cleaner-Lite/3000-2248_4-10398069.html
After installing it, point it to your My Music directory, in just a few seconds it will tell you which files are 0 length files, which files are exactly identical (full clones), which files have the same content (though not the same name); which files have the same names and sizes (not the same as identical content); which files have the same size, and lastly which files have the same name. You may even want to run this, before rescanning into MusicBee.
After this, MusicBee should be able to keep it organized, as long as the new songs that you add you tell them where in My Music to download them to, instead of allowing your browser to decide where to download them to, which could be anywhere.