Over the past decade, memristive biosensors have demonstrated remarkable capabilities in biological detection. However, a notable limitation has been the loss of the memristive effect during biosensing operations. In this study, we introduce a memristive biosensor that successfully integrates resistive switching behavior with biosensing functionality, for the detection of Prostate-Specific Antigen (PSA). The proposed device incorporates dual Schottky contacts and stacked Silicon Nanowires (SiNWs), which act as biomolecule binding sites. Upon bio-functionalization, a distinct voltage difference-termed the Voltage Gap (Vg)-emerges between current minima during forward and backward voltage sweeps in Current-Voltage (I-V) characteristics, serving as a reliable indicator of target molecule binding. Crucially, the device retains its resistive switching properties even during molecular sensing, addressing a key challenge in existing designs. To explain the electrical behavior of this dual-functional memristive biosensor, we developed a capacitively coupled memristive model. The close alignment of simulation results with experimental data provides valuable insights for optimizing the design and performance of memristive biosensors. These advancements highlight the device's potential for a wide range of biomedical applications, aiming to investigate the feasibility of simultaneously integrating biosensing and memristive switching, thereby paving the way for advanced applications such as in-sensor computing and in-memory sensing.