Due to their potential as excellent dual-comb light sources, the dynamics of dual-color solitons within a single cavity have garnered widespread attention. In this work, we constructed a single-cavity ytterbium-doped passively mode-locked fiber laser (MLFL). By utilizing the birefringent filtering effect, dual-color solitons were successfully generated, and both collision and oscillatory dynamics of the dual-color solitons were observed. Using time-stretched dispersive Fourier transform (DFT), we revealed the details of these two distinct dynamical behaviors. We verified through numerical simulations that by adjusting the modulation depth and bandwidth of the filtering effect, it is possible to switch between the two distinct soliton dynamics behaviors. We believe that our research will contribute to the development of single-cavity dual-comb light sources and the exploration of nonlinear dynamics.