Housing in Amsterdam is (horribly) expensive; rooms and small apartments are also wanted by students and September is a bad month to find a room there.
I would recommend you to look a bit south of the airport along the railroad: Hoofddorp, Nieuw-Vennep or Sassenheim. The city of Leiden is also a university town where students take up a significant part of the available rooms.
And yes, many of the independent intermediates that specialize in rental of individual rooms are scammers, but real estate agents ('Makelaars' in Dutch) may also offer temporary rental of rooms (or apartments) and are generally reliable.