Multi-access Edge Computing (MEC) enables users to handle resource-intensive and latency-sensitive tasks. However, the offloading behaviors, which are closely correlated with wireless channel conditions, can inadvertently reveal users' location information to untrustworthy MEC servers. Existing location privacy-aware task offloading (LPTO) mechanisms have not fully considered and comprehensively analyzed personalized location privacy protection requirements. To address this gap, this paper proposes a differential privacy (DP)-based personalized LPTO mechanism for MEC environments that jointly optimizes the perturbation region, privacy budget, and offloading rate while maximizing the offloading utility. We quantify personalized privacy requirements by incorporating task sensitivity, user privacy preference, and task priority. Then, we propose a two-timescale (2Ts) optimization framework to solve the complex personalized location privacy-aware task offloading optimization problem. Specifically, we optimize the perturbation region on a long timescale to align with long-term privacy requirements. In contrast, the offloading ratio and privacy budget are dynamically optimized on a short timescale based on instantaneous channel states and offloading workloads. Furthermore, we model the privacy-aware offloading problem as a Markov decision process (MDP) and develop a dual-agent deep reinforcement learning (DRL)-based personalized LPTO mechanism (DDPLM) to optimize strategies under dynamic MEC systems. Simulation results validate that the proposed DDPLM achieves personalized location privacy protection while reducing computational costs.