Continuum robot arms, with their hyper-redundant continuously deformable bodies, show great promise in applications deemed impossible for traditional rigid robot arms with discrete links and joints, such as navigating tight corners without getting stuck. However, existing continuum robots suffer from excessive twisting when subjected to offset loading, even resulting from their own body weight, which reduces their dexterity and precision. In this work, we present a continuum manipulator that is capable of providing passive torsional stiffness through an origami-inspired modular design, remedying the non-controllable twist typically present in continuum robots. Our proposed origami continuum module is ~73 times stronger in torsion compared with similar-size continuum modules made out of silicone rubber, while being 50% lighter, and capable of 125% change in length. Building on these physical capabilities, we present an optimization-based method to solve for the inverse kinematics of our multi-segment origami continuum manipulator that ensures smooth motion to follow desired end-effector paths, minimizing vibrations of the long and slender body. Further, taking advantage of the length-change capabilities of our origami manipulator, we devise and evaluate grow-to-shape algorithms to plan for full-body robot insertion motions that follow tortuous paths. Lastly, we showcase various applications of our proposed continuum robot for pick-and-place, inspection/exploration, and robotic art. Our study presents a highly capable continuum robot for safe manipulation and structure inspection applications, with potential for real-world deployment.