Reconfigurable surface acoustic wave (SAW) phase shifters have garnered significant attention owing to their potential applications in emerging fields such as secure wireless communication, adaptable signal processing, and intelligent sensing systems. Among various modulation methods, employing gate voltage-controlled tuning methodologies that leverage acoustoelectric interactions has proven to be an efficient modulation approach that requires a low bias voltage. However, current acoustoelectric devices suffer from limited tunability, intricate heterogeneous structures, and complex manufacturing processes, all of which impede their practical applications. In this study, we present a novel material system for voltage-tunable SAW phase shifters. This system incorporates an atomic layer deposition ZnO thin-film transistors on LiNbO₃ structure. This structure combines the benefits of LiNbO₃'s high electromechanical coupling coefficient (K²) and ZnO's superior conductivity adjustability. Besides, the device possesses a simplified structural configuration, which is easy to fabricate. Devices with different mesa lengths were fabricated and measured, and two of the different modes were compared. The results indicate that both the maximum phase shift and attenuation of the Rayleigh mode and longitudinal leaky SAW (LLSAW) increase proportionally with mesa length. Furthermore, LLSAW with larger effective electromechanical coupling coefficients ($K_{eff}^2$) values exhibits greater phase velocity shifts and attenuation coefficients, with a maximum phase velocity tuning of 1.22% achieved. It is anticipated that the proposed devices will find utility in a variety of applications necessitating tunable acoustic components.