The only help I can add is to maybe turn the volume all the way down before turning off. i use my aux input all the time and if the wire is plugged in and i turn the car on- still in aux mode i hear what you seem to hear. i turn my volume down to avoid this. also i've noticed i'll get some static when i'm listening to music on my phone and its charging at the same time.
weird it happened turning it off when on fm- what happened after that? did the radio come back on? or is it only when u start the car?