The shaft is already dried out ...,,,,
You will never get the shaft dryer then what the humidity is where you live .
I use denatured alcohol , it removes some of the chalk and dirt , grim off the shaft .
Pretty tough or imo impossible to get 100% of the chalk dirt out of the maple wood pores using any cleaners .
Part of the dirt grim is oils from your hands ..
For me I don't care if the shaft looks like it been used or not .
I only clean the shaft so I can make the shaft slick , not clean.
My next step after denatured alcohol is I use lighter fluid and lots of it .
I have a lathe and will spin the shaft and wipe the shaft down with a white paper towel soaked with lighter fluid.
Normally I spend about 5 or 10 minutes trying to get as much dirt grim and chalk out of the shafts wood pores using alcohol and lighter fluid .
Their has been some discussion on why lighter fluid does a great job cleaning protecting and making a shaft slick .
Lighter fluid has Paraffin oil in it , not allot but it might be part of the reason it works good .
Then I wax the shaft with some kind of automotive paste wax ,
I have done some testing/ researching or experimenting with different waxes .
All of the Carnauba car paste waxes all seemed to work ........almost equally IMO .
Allot of people in here like Renaissance wax including me .
I also like the Pure Polymer synthetic wax which has the higher melting point and harder surface then any of the natural waxes .
Synthetic waxes dry slower or not at all if put on to thick .
Paste waxes need to be applied in thin coats .........
Natural waxes are Paraffin ,Bees and Carnauba wax .
Carnauba being the hardest and having the highest melting point which is about 185 degrees F
Please note ,, I don't use any sand paper or anything that will remove wood ..
Alcohol , lighter fluid , wax and burnish or buff ..
Good luck .