The wireless networks beyond the fifth generation (5G) are envisioned to be the platform that will support a vast amount of diversified data-driven applications with stringent requirements in terms of computational accuracy, delay, and energy efficiency. The fulfillment of this objective can be achieved by the convergence of communication and computing networks, enabling the exploitation of edge computing resources and the joint orchestration of the corresponding resources. Mobile edge computing (MEC), which refers to the use of edge serves for offloading tasks from mobile devices, is a particularly promising approach to provide the required computational performance for emerging internet-of-things applications, such as the smart grids, smart industry, healthcare, and smart farming. In this work, we propose the use of an advanced multiple access technique and its joint design with adaptive task offloading, in order to reduce delay and energy consumption. More specifically, the use of generalized hybrid orthogonal/non-orthogonal multiple access (OMA/NOMA) for MEC is introduced, which is theoretically superior to other alternatives from the existing literature. In more detail, the proposed scheme is based on the joint utilization of dynamic user scheduling among OMA/NOMA phases and variable decoding order during the successive interference cancellation in NOMA phase. Also, the system’s orchestration is optimized for both full and partial task offloading. Specifically, in full offloading scenario, the user scheduling, time allocation, and power control are jointly optimized. Regarding partial offloading, the computational resources, i.e., the clock speed of the local processors and the number of offloaded bits, are jointly optimized with the communication resources, taking into account the constraint of the energy that is consumed for both local processing and task offloading, which is particularly challenging due to the non-convex nature of the corresponding optimization problem. All optimization problems are efficiently solved by either using closed-form solutions that provide useful insights or low-complexity algorithms. Finally, simulation results demonstrate the effectiveness of the proposed techniques and provide useful insights on the system’s performance, in terms of average delay and energy consumption.