When I was a kid you could purchase codeine cough medicine OTC in my U.S. state (it was same deal as you describe for Australia -- the patient would have to sign a register so the pharmacist could make sure you weren't buying five bottles a day or whatever).

Nothing works as well for a nagging cough as codeine and terpin hydrate. Nothing.

Unfortunately, you've got to go to a doctor to get it nowadays.

Instead, most people use OTC stuff with DXM (which doesn't work as well, is an abusable drug in its own right, and is quite likely more toxic than the codeine).