Movable antenna (MA) has shown significant potential for improving the performance of integrated sensing and communication (ISAC) systems. In this paper, we model an MA-aided ISAC system operating in a communication full-duplex mono-static sensing framework. The self-interference channel is modeled as a function of the antenna position vectors under the near-field channel condition. We develop an optimization problem to maximize the weighted sum of downlink and uplink communication rates alongside the mutual information relevant to the sensing task. To address this highly non-convex problem, we employ the fractional programming (FP) method and propose an alternating optimization (AO)-based algorithm that jointly optimizes the beamforming, user power allocation, and antenna positions at the transceivers. Given the sensitivity of the AO-based algorithm to the initial antenna positions, a PSO-based algorithm is proposed to explore superior sub-optimal antenna positions within the feasible region. Numerical results indicate that the proposed algorithms enable the MA system to effectively leverage the antenna position flexibility for accurate beamforming in a complex ISAC scenario. This enhances the system's self-interference cancellation (SIC) capabilities and markedly improves its overall performance and reliability compared to conventional fixed-position antenna designs.