Adaptive optics (AO) can efficiently compensate for turbulence-induced distortion in orbital angular momentum (OAM)-based optical wireless communication (OWC) systems. In this paper, we design a modified phase diversity algorithm (MPDA)-based wavefront sensor to enhance the reconstruction accuracy of distorted OAM wavefront information. Aiming to further strike a compelling trade-off between AO system complexity and compensation accuracy, we first construct a novel AO system that applies a quickly and electronically controlled focus-tunable lens (FTL). It decontaminates distorted OAM signaling beams while having a low systemic complexity and superior convergence performance. Furthermore, we propose the 3-modified phase diversity algorithm (3-MPDA) AO scheme relying upon a Fourier intensity and two defocused intensities as the prior information, which beneficially balances the compensation effect and the number of defocused intensities and exhibits good noise robustness against charge-coupled device (CCD) detectors. In summary, this paper provides new insight for designing AO schemes with high compensation performance in communication links.