This research work introduces a new strategy to enhance computation offloading in wireless-powered Edge-Enabled Mobile Computing (EEMC) networks by utilizing Dueling Double Deep Q-Networks with Attention (Dueling DDQN-A). EEMC facilitates the offloading of computational tasks from mobile devices to local edge servers, leading to decreased latency and improved energy efficiency. However, the dynamic and stochastic nature of wireless environments presents significant challenges for real-time task offloading decisions. To address this, we enhance the Dueling DDQN framework by integrating an attention mechanism to prioritize the most relevant features in the state space. The proposed Dueling DDQN-A model enhances decision-making by separating state value estimation from action advantage estimation. Additionally, the model uses attention to prioritize key state features, leading to improved adaptability in dynamic wireless channel and network environments. We frame the offloading decision problem as a Markov Decision Process (MDP) and apply deep reinforcement learning to optimize both the computation rate and energy efficiency. We compare and evaluate the model through comprehensive simulations using the latest techniques like Simple DQN, Double DQN, and Dueling DDQN. Our results demonstrate that Dueling DDQN-A achieves a 59.12% improvement in average computation rate and a 16.19% reduction in energy consumption over the baseline models. Additionally, it significantly outperforms the baselines in terms of training loss and convergence speed. These findings suggest that Dueling DDQN-A offers a robust and highly efficient solution for real-time computation offloading in wireless-powered EEMC networks, making it a strong candidate for deployment in real-world scenarios.