When you have a movieclip in the library it can be set to export for actionscript, and you would put in the location of an external Class file. If you made your current scene be a movieclip, and in the library you get the properties for the movieclip, in Advanced view you would select the export for actionscript box, and for the Class file you would enter the location of your current document class. Then you could place that movieclip on the stage and start with a new document class for the FLA.
It would make sense to change some names though. For example, if your FLA is named MyGame, and your current document class is MyGame.as, you probably want to keep that the same, for the new fresh document class. So, for the new movieclip you would change the two places that say MyClass to perhaps MyClassScene1, and save that copy of the file as MyClassScene1.as. In the library movieclip properties I mentioned, you would use MyClassScene1 as the class name.
Finally, in the new FLA document class you no doubt will want to keep some of the global level functions, and remove all the scene specific functions, and in the duplicate renamed MyClassScene1.as you would keep all the scene specific functions and delete any of the global functions that will now be handled by the new document class.