In this paper, we present a novel synergistic framework for learning shape estimation and a shape-aware whole-body control policy for tendon-driven continuum robots. Our approach leverages the interaction between two Augmented Neural Ordinary Differential Equations (ANODEs) -- the Shape-NODE and Control-NODE -- to achieve continuous shape estimation and shape-aware control. The Shape-NODE integrates prior knowledge from Cosserat rod theory, allowing it to adapt and account for model mismatches, while the Control-NODE uses this shape information to optimize a whole-body control policy, trained in a Model Predictive Control (MPC) fashion. This unified framework effectively overcomes limitations of existing data-driven methods, such as poor shape awareness and challenges in capturing complex nonlinear dynamics. Extensive evaluations in both simulation and real-world environments demonstrate the framework's robust performance in shape estimation, trajectory tracking, and obstacle avoidance. The proposed method consistently outperforms state-of-the-art end-to-end, Neural-ODE, and Recurrent Neural Network (RNN) models, particularly in terms of tracking accuracy and generalization capabilities.