This is how I clean my shafts, as I don't have a lathe.
I use a dry magic eraser, (wet w/ denatured alcohol if real dirty)
Then I use a piece of leather to burnish it.
Then I apply a coat or two of Butchers Bowling Alley Wax for a slick finish.
There is no need for a lathe to clean a shaft with magic eraser and DA. The point is to remove chalk and grime from the grain, which is not circular around the shaft, so clean it with the grain. It's much easier. Also, the DA and magic erasor will remove the sealer and wax to a degree at least, depending on how dirty the shaft was of course, and so it will feel rougher because now the grain has been opened, so after cleaning seal it well, let dry, then sand the sealer, not the wood. You can tell by the removed material on the sandpaper, if it's wood, the sandpaper is too course. Once it is smooth as you can get it with whatever sand paper you use,,,,,, I like steps from 600 to 4k,,,,,,,,, then wax it, and burnish the wax last. To burnish you should feel some heat in the burnish cloth, or leather,,,,,, this is important because it evaporates the petrol products from the wax faster, so the wax can harden faster. Heat, but not hot.
Make sure the shaft is dry before applying wax and all the eraser chemicals are gone. Leather works for burnishing or a sheet of paper.