This paper addresses the NP-hard problem of optimizing container handling at ports by integrating Quay Crane Dual-Cycling (QCDC) and dockyard rehandle minimization. We realized that there are interdependencies between the unloading sequence of QCDC and the dockyard plan and propose the Quay Crane Dual Cycle - Dockyard Rehandle Genetic Algorithm (QCDC-DR-GA), a hybrid Genetic Algorithm (GA) that holistically optimizes both aspects: maximizing the number of Dual Cycles (DCs) and minimizing the number of dockyard rehandles. QCDC-DR-GA employs specialized crossover and mutation strategies. Extensive experiments on various ship sizes demonstrate that QCDC-DR-GA reduces total operation time by 15-20% for large ships compared to existing methods. Statistical validation via two-tailed paired t-tests confirms significant improvements at a 5% significance level. The results underscore the inefficiency of isolated optimization and highlight the critical need for integrated algorithms in port operations. This approach increases resource utilization and operational efficiency, offering a cost-effective solution for ports to decrease turnaround times without infrastructure investments.