Verb order — which way is right?
I think either version is acceptable and the meaning is clear.
You don't need softly at the end.
"On the radio" instead of "Over the airwaves" either is correct, but Over the airwaves might be considered old fashioned, so if you were trying to invoke a retro atmosphere it could be more appropriate.
It is only the first movement of the Moonlight Sonata that is slow and melancholy (Adagio sostenuto). The second movement is Allegretto and the third Presto agitato. So you could say " ... played the melancholy piano chords from the opening movement of the Moonlight Sonata."