One thing I learned many years ago was to use your fingers of your left hand and continue your tool work with your right hand. Using your fingers seems to be a bit hard on your fingers, but it's not if you use a triple thickness of bounty paper towels, or a small doubled cloth. A little practice helps.
I make many bic type pens for give-away. If you are getting a LOT of flex, you neen to use a harder wood. Coco Bola comes to mind. There are others......I simply will not turn maple as it is too hard. You have to put too much pressure on Maple nd it will chatter until it breaks. I also make many hair sticks which I consign to Beauty shops. These are turned to a much smaller diameter, but the towels help, especially using a skew.
I hope this works for you