To save the current configuration into a separate file, use the "Save a Copy As…" menu item in the File menu. This will default to saving the file in the "DragThing Settings" folder, the contents of which appear in the Setting menu. A folder of settings files placed in the "DragThing Settings" folder will appear as a hierarchical menu in the Settings menu and so on.
You can open the "DragThing Settings" folder in the Finder by selecting the "Open Settings Folder" item in the Settings menu. It's not necessary to store settings files in the settings folder however.
Selecting the item from the Settings menu will close and backup the current settings file, and open up the new one. You can also drag a settings file to a dock and double-click it to switch, or double-click it from the Finder.
DragThing will create a "DragThing Startup Settings" alias in the Preferences folder, pointing to the settings file to use at startup, although double-clicking a file while DragThing is not running will override this.