The AABA form only applies to the chorus. The whole song typically began with a short intro, then a "verse" that often set the scene. Then came the chorus which was the catchy part of the song and usually the only part anyone remembers. Finally there would be however much other stuff as was needed to fill out the performance and end with a coda.

"Ja Da" from 1918 is a textbook example: https://www.youtube.com/watch?v=q1-oGzXzNHo