So anyhoo
I went to Project Setup from the Project Menu. Selected Sample Rate and changed it to 48k. I then changed the Bit Resolution from 24 to 32. Lo and behold, all audio files now play at the correct pitch! No more Pinky and Perky.
Not sure if the other steps contributed to the correction or if the sample rate and bit resolution change did it. But hey, it works!
Just a pity I can’t give myself a ‘Thanks.’ 
Just is case somebody else has this problem in the future, here are some headwords that might search guide you here:
Altered pitch, unwanted transposition, sharpened playback. Have fun!