This will probably sound crazy, but I don't have to really clean my shaft.
I seal it often by rubbing it over with paper until it gets really warm. If there is any dirt on the shaft, it will be on the paper afterwards. This is mainly oil from my finger prints, visible as a grey residue on the paper.
From time to time, I will use furniture polish (it's a product from Johnson's, though apparently not sold in the US) or cue silk on the shaft, but mainly I just rub it with paper about once a week.
The result is that the shafts are so slick that they don't absorb chalk well.
And: Using other colours of chalk than blue seems to help a lot
I have used mainly green chalk on my Kruger, and use grey chalk nowadays. Tweeten claims that there is no real difference, and I find no real difference. Moisture seems to make a difference, but not colour. If a piece of chalk does not work well, I give it some drops of water and let it dry over night.
Regards,
Detlev