In this paper, we propose a dynamic model and control framework for tendon-driven continuum robots (TDCRs) with multiple segments and an arbitrary number of tendons per segment. Our approach leverages the Clarke transform, the Euler-Lagrange formalism, and the piecewise constant curvature assumption to formulate a dynamic model on a two-dimensional manifold embedded in the joint space that inherently satisfies tendon constraints. We present linear and constraint-informed controllers that operate directly on this manifold, along with practical methods for preventing negative tendon forces without compromising control fidelity. This opens up new design possibilities for overactuated TDCRs with improved force distribution and stiffness without increasing controller complexity. We validate these approaches in simulation and on a physical prototype with one segment and five tendons, demonstrating accurate dynamic behavior and robust trajectory tracking under real-time conditions.