As a fundamental component of a linac-based beam injector, the rectangular-waveguide coupler is a conventional device for feeding high power, but it will induce field asymmetry and resonant-frequency shifting. Furthermore, it is also difficult to adjust the coupling factor for adjustable beam injectors. In this paper, an equivalent circuit model is established for the coupler with a tuning rod. Based on theoretical analysis, the optimal position for the rod is given. Besides, the frequency shifting is corrected by using another rod inserted to the cavity in the opposite direction. Sizes and adjustment ranges of both rods are given by three-dimensional electromagnetic simulation using CST MICROWAVE STUDIO. Jointly adjusted simulation results show that, critical-coupled states are achieved for different beam intensity while the resonant frequency remains stable, thus the risk of reflected power caused by coupler mismatching can be avoided, and the field asymmetry due to a small coupling hole can be reduced.