Understanding elastic instability has been a recent focus of concentric tube robot research. Modeling advances have enabled prediction of when instabilities will occur and produced metrics for the stability of the robot during use. In this paper, we show how these metrics can be used to resolve redundancy to avoid elastic instability, opening the door for the practical use of higher curvature designs than have previously been possible. We demonstrate the effectiveness of the approach using a three-tube robot that is stabilized by redundancy resolution when following trajectories that would otherwise result in elastic instabilities. We also show that it is stabilized when teleoperated in ways that otherwise produce elastic instabilities. Lastly, we show that the redundancy resolution framework presented here can be applied to other control objectives useful for surgical robots, such as maximizing or minimizing compliance in desired directions.