The next-generation mobile network anticipates integrated heterogeneous fronthaul and backhaul technologies referred to as a unified crosshaul architecture. The crosshaul enables a flexible and cost-efficient infrastructure for handling mobile data tsunami from dense Internet of things (IoT). However, stabilization, energy efficiency, and latency have not been jointly considered in the optimization of crosshaul performance. To overcome these issues, we propose an orchestration scheme referred to as the stabilized green crosshaul orchestration (SGCO). SGCO utilizes a Lyapunov-theory-based drift-plus-penalty policy to determine the optimal amount of offloaded data that should be processed either at the eastbound or westbound computing platforms to minimize energy consumption. To achieve system stability, the cache buffer is considered as the main constraint in developing the optimization process. Moreover, the amount of offloaded data transmitted via crasshaul links is selected by adopting the binary min-knapsack problem. Accordingly, a lightweight heuristic algorithm is proposed. As the cache buffer is stabilized and the computations are controlled, the SGCO ensures adjustable computing latency threshold for various IoT services. The performance analysis shows that the proposed SGCO scheme exposes effective energy consumption compared to other existing schemes while maintaining system stability considering latency.