It's tough to tell you exactly what you need to do differently unless a person is right there next to you but here are a few tips...
1) Not too much tailstock pressure. Too much will compress the whole thing and make it whip more. Just enough pressure to keep the wood against your drive center.
2) What kind of tool are you using? And how are you using it? Tough to describe online how to do it but a light tough at a shearing angle towards the headstock is best. If any project demands the use of a skew, this is the one.
3) Sharp tools with correct bevels and proper technique. I know, this is a throw-away sentence as it doesn't tell you much. It's only when you find out that what you have been doing isn't optimal, that you really know how meaningful that sentence is.
4) Good job at providing a steady with your other fingers. Are they as close to the tool (on the other side of course) as possible? Choke up on the tool in your tool-holding hand. Gives you more control.
5) Are you working with green wood? Green wood will whip far more than dry at a given diameter. What kind of wood are you using? How long is it?
6) Don't push too hard at all. Light shearing cuts almost parallel with the axis of the lathe (that's the scientific phrase for cutting toward either the headstock (preferred) or tailstock and not straight at the wood).
7) Part off at the headstock end! Part off with a skew or small spindle gouge at a shearing angle. Don't use a "small skew in a peeling cut" (aka Parting Tool) here. This is, I assume, how you're doing it now?
If you'll email me direct, we can setup a time in a day or two to show you how I do it through the LatheCAM. Might be easier to see it done live and (sort-of) in person.
Small, thin and long turning is a very demanding turning project. The smaller, thinner and longer you get, the more challenging and technical.
- Andrew