Safe, smooth, and optimal motion planning for nonholonomically constrained mobile robots and autonomous vehicles is essential for achieving reliable, seamless, and efficient autonomy in logistics, mobility, and service industries. In many such application settings, nonholonomic robots, like unicycles with restricted motion, require precise planning and control of both translational and orientational motion to approach specific locations in a designated orientation, such as for approaching changing, parking, and loading areas. In this paper, we introduce a new dual-headway unicycle pose control method by leveraging an adaptively placed headway point in front of the unicycle pose and a tailway point behind the goal pose. In summary, the unicycle robot continuously follows its headway point, which chases the tailway point of the goal pose and the asymptotic motion of the tailway point towards the goal position guides the unicycle robot to approach the goal location with the correct orientation. The simple and intuitive geometric construction of dual-headway unicycle pose control enables an explicit convex feedback motion prediction bound on the closed-loop unicycle motion trajectory for fast and accurate safety verification. We present an application of dual-headway unicycle control for optimal sampling-based motion planning around obstacles. In numerical simulations, we show that optimal unicycle motion planning using dual-headway translation and orientation distances significantly outperforms Euclidean translation and cosine orientation distances in generating smooth motion with minimal travel and turning effort.