At the end of the last of my three lessons LB asked to look at my melodeon - saying that it sounded a bit 'wheezy'. Investigation showed a long split where the bellows had parted company from the plastic bit (no idea what this is called) which plugs into the wooden ends of the melodeon. It was easily fixed but LB said that he would need to keep the bellows in a press overnight for the glue to set.
I went back the next day to find my melodeon smelling strongly of Evostik - and working far better. A lot of the problems that I was having with running out of air just vanished. I've now got much less need to use the air button, and where I am using it it tends to be at very different times to previously. Some tunes that we had marked up as needing air at certain points just don't need the air button at all now.
EDITED to add:
Suddenly I'm finding it easier - I don't have wheezy bellows any more (goodness knows how long that problem was there). I've lost my fear of moving my hand about the keyboard, and everything is much easier. I'm even trying to do some bass with some of the tunes. I'm really starting to feel like I'm making some progress now.