Felix, thanks for your reply.
Yes, I have worked my way through the thread you mentioned, though I must admit I did not absorb it all and don't recall seeing a specific conclusion of a hardware fault but there is a good chance I missed it. Anyway I have bought quite a few of these modules so I would like to continue to experiment if for nothing else other than a learning exercise.
I understand that the listen mode burst is offset as per the code you referenced. Just to be clear, I am using the current RFM69 library without any modification. My question relates to the fact that the module seems to be transmitting at both the offset frequency of 919MHz as well as the normal frequency of 915MHz at the same time (as per the screen shot in my original post). Now this may be caused harmonics or reflections or bad antenna design or proximity between transmitter and receiver or whatever rather than by the module itself, but I have a very limited understanding of RF so I can't really tell. I just thought it looked a bit strange and may have been causing some of the reliability issues I have been experiencing.
I am not ready to give up on Listen Mode yet as I can see it would be handy in my application so I will continue to experiment/play.