With the rapid development of global satellite Internet and the growing demand for seamless connectivity, the Satellite–Terrestrial Integrated Network (STIN) has become a key architecture for achieving ubiquitous communication coverage. However, STIN faces critical energy efficiency challenges, including the high peak-to-average power ratio (PAPR) that degrades high power amplifier efficiency at the physical layer and frequent satellite handovers that increase network overhead. To address these issues, this paper proposes a mobility-aware model of Multicarrier Direct-Sequence Spread Spectrum (MC-DSSS) STIN. At the physical layer, the Orthogonality-Based Generalized Multicarrier Constant Envelope Multiplexing (CEMIC) technique is adopted, and an energy efficiency maximization problem is formulated under constant-envelope constraints. A joint power allocation algorithm is developed based on the Dinkelbach method and the Alternating Direction Method of Multipliers (ADMM) to solve the non-convex problem efficiently. To overcome the limitations of single-layer optimization, a twolayer collaborative framework is further proposed. At the network layer, an improved binary particle swarm optimization (IBPSO-HO) algorithm is employed to optimize satellite handovers. This joint design enables two-layer energy efficiency optimization. Simulation results demonstrate that the proposed scheme significantly enhances overall energy efficiency across both layers, providing robust theoretical support for the large-scale green deployment of STIN.