In this paper, we explore the utilization of both full-duplex (FD) and half-duplex (HD) transmission modes on unmanned aerial vehicles (UAVs) to enhance wireless system performance. We consider two practical scenarios: one without a direct transmitter-user link and the other with such a link. For both cases, we derive mathematical expressions of outage probabilities (OPs) and throughputs of FD-UAV and HD-UAV systems, employing a realistic channel model aligned with the fifth and beyond generations (5G-B5G) standards. In cases where imperfect self-interference cancellation (SIC) occurs in FD-UAV systems, we introduce an optimal power allocation approach to enhance system performance. Numerical results underscore the benefits of cooperative communications, particularly when combining the direct link with the UAV link at the user, leading to an overall enhancement in system performance. Furthermore, we conduct a comprehensive analysis of various system parameters, including predefined rates, residual self-interference (RSI) levels, high carrier frequencies of Wi-Fi networks, and high altitudes of UAVs. The impact of RSI is particularly notable, and the proposed optimal power allocation approach significantly improves system performance in such cases. Specifically, the introduced scheme helps avoid error floors in regions of high transmit power, enabling throughputs to reach desired targets in FD-UAV systems. Crucially, the optimal power value is considerably lower than the traditional value often used without optimal power allocation, extending the operational duration of FD-UAV. Finally, to validate the derived mathematical expressions and affirm the effectiveness of the proposed approaches, we conduct Monte-Carlo simulations.