How can I create a modifier shortcut to drag files into a new folder?
As of OS X 10.9 “Mavericks”, you can do this right from the Finder:


Select your files.
Drag ’n’ drop the files to their new destination.
Press ⌃⌘N to create a “New Folder with Selection”.
Optionally, type a new name the folder and press return (otherwise, the file will be named “New Folder With Items”).


You bring up 3rd-party apps. I don’t see how Keyboard Maestro could help in this particular instance.  Hazel — or indeed Apple’s own folder actions — might, but only if your target was always the same folder or set of folders.

But you might want to look at some of LaunchBar’s less known capabilities: here’s a deep navigationless link to their keyboard-powered drag-and-drop instructions.