hit from the 60s girl-group genre:

Yes, but I really should have been more explicit about the type of song I was thinking about. The song Dusty by The Rag Dolls had a certain unnerving quality about it. It didn't have the lyrics of a sad tale, one that you liked even though it brought a tear to your eye. Neither was it of the upbeat variety in the style of Peggy March or The Secrets with their Boy Next Door hit. Instead, after listening to Dusty and similar creepy tunes you (or perhaps I) would experience a mildly disheartening sensation, of a sort like having your soul encased in pustulent ooze. After listening to Dusty I feel a need to shower off the spiritual slime that encases my psyche. However, I know that no matter how many times I shower the essence of Dusty will never depart. 

As I said in the OP, for no comprehensible reason this song (Dusty) entered my thoughts. It was a disturbingly creepy tune but it's embedded in my memory. Here are others of that sort: