Convex free regions provide a structured and optimization-friendly representation of collision-free space for robot navigation in unknown and cluttered environments. However, existing methods typically enlarge local collision-free regions mainly according to surrounding obstacle geometry. In cluttered environments, such strategies may fail to generate regions that both accommodate robot geometry and preserve traversable extension along candidate motion directions, thereby limiting downstream traversal, especially in narrow passages. Even when such a region is available, safe motion generation remains challenging, because safety checking at discretized trajectory samples does not guarantee continuously collision-free motion when robot geometry is modeled explicitly. To address these issues, we propose a navigation framework that jointly incorporates candidate motion directions and robot geometry into convex free-region generation, and achieves continuously collision-free motion through continuous-safe trajectory generation. Within each region, the framework performs geometry-aware target pose selection and trajectory generation, together with Lipschitz-based continuous safety certification and local refinement. The resulting free regions and candidate motions are maintained in a region-based graph to support incremental planning. Quantitative results in cluttered 2D navigation scenarios show that the proposed method generates free regions better aligned with downstream traversal and enables reliable collision-free navigation, while additional 3D and real-world experiments on a quadrupedal robot and a UAV demonstrate the extensibility and practical applicability of the framework. The open-source project can be found at https://github.com/ZhichengSong6/FRGraph.