We present an efficient algorithm for motion planning and control of a robot system with a high number of degrees-of-freedom. These include high-DOF soft robots or an articulated robot interacting with a deformable environment. Our approach takes into account dynamics constraints and present a novel technique to accelerate the forward dynamic computation using a data-driven method. We precompute the forward dynamic function of the robot system on a hierarchical adaptive grid. Furthermore, we exploit the properties of underactuated robot systems and perform these computations for a few DOFs. We provide error bounds for our approximate forward dynamics computation and use our approach for optimization-based motion planning and reinforcement-learning-based feedback control. Our formulation is used for motion planning of two high DOF robot systems: a high-DOF line-actuated elastic robot arm and an underwater swimming robot operating in water. As compared to prior techniques based on exact dynamic function computation, we observe one to two orders of magnitude improvement in performance.