" But you still haven't explained how to get rid of the huge folder of ALL the pictures, that are now divided up into smaller folders......(of the same pics) "
After having "moved" all the photos to other (smaller) folders, the big one should be empty !

"If I delete the albums, will they still be online for family to see?"
If you did not "Sync to web", then yes, the webalbums remain where they were.
Just be careful and delete one photo and check. If that's OK then you can go on.