Digital twins (DTs) have emerged as a promising enabler for representing the real-time states of physical worlds and realizing self-sustaining systems. In practice, DTs of physical devices, such as mobile users (MUs), are commonly deployed in multi-access edge computing (MEC) networks for the sake of reducing latency. To ensure the accuracy and fidelity of DTs, it is essential for MUs to regularly synchronize their status with their DTs. However, MU mobility introduces significant challenges to DT synchronization. Firstly, MU mobility triggers DT migration which could cause synchronization failures. Secondly, MUs require frequent synchronization with their DTs to ensure DT fidelity. Nonetheless, DT migration among MEC servers, caused by MU mobility, may occur infrequently. Accordingly, we propose a two-timescale DT synchronization and migration framework with reliability consideration by establishing a non-convex stochastic problem to minimize the long-term average energy consumption of MUs. We use Lyapunov theory to convert the reliability constraints and reformulate the new problem as a partially observable Markov decision-making process (POMDP). Furthermore, we develop a heterogeneous agent proximal policy optimization with Beta distribution (Beta-HAPPO) method to solve it. Numerical results show that our proposed Beta-HAPPO method achieves significant improvements in energy savings when compared with other benchmarks.