This study departs from the prevailing assumption of independent Transmission and Reflection Coefficients (TRC) in Airborne Simultaneous Transmit and Reflect Reconfigurable Intelligent Surface (STAR-RIS) research. Instead, we explore a novel multi-user downlink communication system that leverages a UAV-mounted STAR-RIS (Aerial-STAR) incorporating a coupled TRC phase shift model. Our key contributions include the joint optimization of UAV trajectory, active beamforming vectors at the base station, and passive RIS TRCs to enhance communication efficiency, while considering UAV energy constraints. We design the TRC as a combination of discrete and continuous actions, and propose a novel Dual Actor Deep Deterministic Policy Gradient (DA-DDPG) algorithm. The algorithm relies on two separate actor networks for high-dimensional hybrid action space. We also propose a novel harmonic mean index (HFI)-based reward function to ensure communication fairness amongst users. For comprehensive analysis, we study the impact of RIS size on UAV aerodynamics showing that it increases drag and energy demand. Simulation results demonstrate that the proposed DA-DDPG algorithm outperforms conventional DDPG and DQN-based solutions by 24% and 97%, respectively, in accumulated reward. Three-dimensional UAV trajectory optimization achieves 28% higher communication efficiency compared to two-dimensional and altitude optimization. The HFI based reward function provides 41% lower QoS denial rates as compared to other benchmarks. The mobile Aerial-STAR system shows superior performance over fixed deployed counterparts, with the coupled phase STAR-RIS outperforming dual Transmit/Reflect RIS and conventional RIS setups. These findings highlight the potential of Aerial-STAR systems and the effectiveness of our proposed DA-DDPG approach in optimizing their performance.