Upgraded Galaxy S III to 4.0.4, can't drag app to taskbar anymore. How can I enable dragging again? I upgraded my Galaxy S III to Android 4.0.4 last night and now I can't drag apps to the taskbar anymore.

Is there a setting I can change to be able to drag apps to taskbar again?
According to the User's Manual, there isn't a setting that locks the Primary Shortcuts.  So I would suggest turning the phone off and then restarting it.

Here are the instructions from the User's Manual for Adding and Removing Primary Shortcuts.  Perhaps there's a step here that you've missed.


  You can change any of your primary shortcuts with the exception of the
  Apps shortcut. The primary shortcuts are the bottom row of app
  shortcuts that remain static on all home screens (by default Phone,
  Contacts, Messaging, Internet, and Apps).   



Note: Before replacing a primary shortcut, you must first add the replacement shortcut to the Home Screen.


  
  Press to display the main Home screen.
  Touch and hold the primary shortcut you want to replace and then drag it to an empty space on any Home screen.
  Scroll through the Home screen to find the shortcut you want to add as the new primary shortcut.
  Touch and hold the new shortcut and then drag it to the empty space in the row of primary shortcuts.
  The new primary shortcut will now appear on all Home screens.