I use a towel after each session and remove as much chalk as I can from the tip before putting it in my case. I found that for any chalk, when you slide the cue into the case, over time you end up with chalk lining the inside of the shaft tube, and eventually, it ends up all over the shaft. This also prevents building up too much chalk if you are an over-chalker.
After trying many of the "big names", I settled on V10. I have never seen "chunks" on my table, and I have Blue 860 cloth so the green would be obvious.