The problem that you describe is common in spindle turning when the turning piece is too flexible. A condition referred to as whipping occurs when you press your too into the workpiece it flexes or slightly bends.
I can't say exactly how stiff a spindle must be in order to avoid whip, but if you can see the piece bend by pressing it with the force that you would apply when cutting, you are in for trouble.
There are ways to keep a spindle from bending. They basically entail bracing the spindle from the back side. You can buy devices called "steady rests" which are used commercially and some form is used by pool cue manufacturers. I just Googled "steady rest lathe" and found that they are apparently available for well under a hundred dollars (US). All three wheels are adjusted to be in contact with the spindle.
From personal experience, I can tell you that a duplicator is not the way to go, unless you also have a steady rest.
When using the steady rest, you will do your cutting where the rest isn't, so you will need to move it from time to time.