Integrating wind energy into power systems can negatively impact stability by reducing oscillation damping. Wind Turbine Voltage Regulators (WT VRs) are designed to manage reactive power and maintain voltage stability; however, they often do not coordinate effectively with Power System Stabilizers (PSS) from synchronous generators (SG). This study utilizes the GOOSE Optimization Algorithm (GOA) to optimize and coordinate the gains of the WT proportional-integral virtual regulator (WT PI-VR) and the SG proportional-integral-type lead-lag PSS (PI-type LL-PSS), to enhance power system stability and performance. The GOA performance compared with the Osprey Optimization Algorithm (OOA) and Particle Swarm Optimizer (PSO). The PI-type LL-PSS performance is compared with proportional-integral-derivative PID-PSS configurations, highlighting its robustness. Testing scenarios include step changes, voltage sags, and three-phase short-circuit faults, using metrics like integral time absolute error, settling time, and standard deviation for robustness evaluation. Statistical analysis shows several benefits from the proposed methodology: (i) A 48.85% stability improvement in coordinating WT PI-VR with PID-PSS using GOA versus OOA, (ii) A 24.40% performance boost with GOA over OOA using PI-type LL-PSS, (iii) A 14.4% enhancement when coordinating WT PI-VR with PI-type LL-PSS compared to PID-PSS, and (iv) A 34.23% performance increase using GOA instead of PSO for coordinating WT PI-VR with PI-type LL-PSS.