Energy-absorbing anchor support is a key technology for controlling large deformations in high-stress surrounding rock. However, the relationship between the geometric parameters of conventional anchor energy-consuming components and their energy-consuming performance is unclear, which limits the optimization of these parameters. This paper designs a cone-head-sleeve damping structure as the core energy-consuming component of energy-absorbing anchors. Optimizing its cone angle (α) and sleeve wall thickness (t) parameters improves anchor support performance. Through indoor experiments and numerical simulations, we reveal the resistance characteristics and energy dissipation law of the damping structure during the slip process and establish a design method for optimizing the parameters. The results show that the SCATW (small cone angle and thick wall) design principle should be followed when using conventional materials to fabricate the damping structure. When the cone angle is between 0.5° and 5° and the sleeve wall thickness is between 4 and 7 mm, the damping structure has greater stabilizing resistance and better energy dissipation performance. A method of optimizing the geometric parameters to enhance energy dissipation is provided for engineering applications. with a 2° cone angle as the neutral point. In other words, when the cone angle α is between 0.5° and 2° or between 2° and 5°, respectively, increasing the cone angle and wall thickness improves the energy dissipation of the damping structure.