This paper presents a shape-aware whole-body control framework for tendon-driven continuum robots with direct application to endoluminal surgical navigation. Endoluminal procedures, such as bronchoscopy, demand precise and safe navigation through tortuous, patient-specific anatomy where conventional tip-only control often leads to wall contact, tissue trauma, or failure to reach distal targets. To address these challenges, our approach combines a physics-informed backbone model with residual learning through an Augmented Neural ODE, enabling accurate shape estimation and efficient Jacobian computation. A sampling-based Model Predictive Path Integral (MPPI) controller leverages this representation to jointly optimize tip tracking, backbone conformance, and obstacle avoidance under actuation constraints. A task manager further enhances adaptability by allowing real-time adjustment of objectives, such as wall clearance or direct advancement, during tele-operation. Extensive simulation studies demonstrate millimeter-level accuracy across diverse scenarios, including trajectory tracking, dynamic obstacle avoidance, and shape-constrained reaching. Real-robot experiments on a bronchoscopy phantom validate the framework, showing improved lumen-following accuracy, reduced wall contacts, and enhanced adaptability compared to joystick-only navigation and existing baselines. These results highlight the potential of the proposed framework to increase safety, reliability, and operator efficiency in minimally invasive endoluminal surgery, with broader applicability to other confined and safety-critical environments.