But is "Dingos eat Babies" a common saying, or is the 1980 event the origin of this?
Correct, the real life event eventually spawned a movie with Tom Selleck and Laura San Giacomo called, "Quigley Down Under". In the movie, where she plays an apparent crazy person for a while, there is the dingo scene. After that scene, the two meet back up and she starts frantically screaming, "The dino ate my baby".
After that, many people who liked to quote movies latched on to the words (and way she said it) and it became a popular saying.