It doesn't matter much if a spindle is wobbles a bit; it can still spin perfectly good yarn. But if it bothers you a lot, check to see that the hook is perfectly upright and doesn't lean to one side or the other. If it does, try bending it the tiniest bit with your fingers or a pliers -- be very gentle; you don't want to break the hook.
This might help reduce the wobbling. But it won't really help with the thick-and-thin yarn you're getting. That's a matter of learning to draft evenly. Go slow and hold the clump of fiber loosely. And (this must be getting boring) practice, practice, practice.
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