if editor.getSelText() == "":
	notepad.menuCommand(MENUCOMMAND.SEARCH_FINDNEXT)
else:
	notepad.menuCommand(MENUCOMMAND.SEARCH_SETANDFINDNEXT)

To bind this to F3, first add the script as Menu item in PythonScript config:

Restart NPP, than in Settings -> Shortcut mapper -> Plugin commands search for your script’s filename:

That’s not the way that the Scintilla IP that Notepad++ uses works. The key will do one thing – either “Find Next” or “Select and Find Next”.

But if you remove the shortcut for Find Next and change Select and Find Next to F3, then it will likely come close to what you want… because when you hit it a second time, whether the old word is still in the buffer for searching, or whether you have selected new text since the last instance of Select and Find Next, when you hit Select and Find Next again, it will usually find what you expect, based on my understanding of your expectations.

Settings > Shortcut Mapper allows you to change shortcuts. As @Stefan-Pendl said, that feature defaults to Ctrl+F3