Multi-access edge computing (MEC) has emerged as a pivotal paradigm to enhance the efficiency of computation offloading by placing computing resources closer to data sources. In this paper, we tackle the challenge of energy consumption and latency in MEC environments while maintaining task deadlines and ensuring reliable communication. We present an integrated optimization approach that dynamically adjusts the modulation index and offloading decisions to minimize energy consumption per bit. Our approach considers task deadlines, application bit error rate (BER) constraints, and system limitations. We propose a distributed offloading decision algorithm that categorizes tasks into strict and loose deadline tasks, optimizing resource allocation between the mobile device and edge server. Furthermore, we employ an earliest deadline first (EDF) scheduling policy for efficient task scheduling. Through extensive simulations, we demonstrate that in certain scenarios, our method achieves significant energy savings of 53.5% while upholding BER requirements and reducing communication delays.