Navigational bronchoscopy is critical for pulmonary interventions, yet current platforms depend heavily on pre-operative CT or external sensors, limiting their use in critical care and resource-constrained settings. Vision-only navigation offers a scalable alternative, but conventional visual odometry (VO) struggles with texture-poor airway images, specularities, and the vanishing-point singularities of tubular anatomy, leading to frequent tracking failures and drift. We present a geometry-aware VO framework that explicitly leverages vanishing-point cues from airway lumens. Detected lumens are back-projected to 3D rays, whose weighted fusion yields a stable forward heading even when parallax cues are absent. This heading, together with looming-based velocity estimates, is fused with noisy VO outputs using a bespoke high-gain observer that enforces airway-following priors and rejects drift. We validate the method on ex-vivo mechanically ventilated human lungs with electromagnetic tracking ground truth. Compared to state-of-the-art pipelines (ORB-SLAM2, LoFTR-VO, DPVO), our approach reduces absolute trajectory error by more than 50% and achieves the lowest relative pose error across all test sequences.