I use a 4 jaw pin chuck so I can grab everything well. I use a lot of electrical tape, i use sandpaper more than any tools, and when I do use tools a little, I use carbide ones at a slow speed.
On a bad day when I start breaking them (because of the thin neck), I quit and go to something else. It takes a patient hand to do it right.
On the wobble, you may want to see if you are turning it with the reed in place while turning. This is a must in my opinion.
Vince