Mobile Edge Computing (MEC) has been proposed to enhance the performance of Internet of Things (IoTs) devices by offloading computation-intensive tasks to nearby edge clouds, while Non-Orthogonal Multiple Access (NOMA) enables multiple IoTs devices to share subcarriers with varying power levels, making it ideal for computation offloading. Despite the potential benefits, integrating NOMA with MEC presents complex challenges, including resource allocation, decision optimization, and balancing energy efficiency with completion time. In this paper, we address the computation offloading and resource allocation problem in NOMA-MEC enabled IoT networks, aiming to minimize completion time and maximize energy efficiency while meeting processing latency requirements. Our model supports partial computation offloading, allowing devices to partition tasks for both local execution and offloading to the edge clouds. To this end, we first introduce two processes, i.e., infeasible tasks elimination and admission control, to improve algorithm efficiency. Then, we propose an iterative algorithm, comprising two low-complexity sub-algorithms, to address various optimization aspects, including CPU frequency allocation, offloading decisions, time allocation, transmit power control, and network resource allocation. Extensive simulations validate that our approach outperforms existing methods in terms of completion time, total saved energy, and offloading ratio.