In this study, a rate-based absorption model coupled with an improved thermodynamic model was developed to characterize the removal of acid components (CO2 and H2S) and organic sulfur (COS and CH3SH) from natural gas with an aqueous sulfolane–MDEA solution. First, the accuracy of the thermodynamic model was validated by comparing the calculated partial pressure of CO2, H2S, and CH3SH with those of the experimental data reported in the literature. Then, the industrial test data were employed to validate the absorption model and the simulation results agreed well with the experimental data. The average relative errors of the removal rates of CO2, COS, and CH3SH are 3.3%, 3.0%, 4.1%, respectively. Based on the validated coupled model, the total mass transfer coefficient and mass transfer resistance of each solute component at different column positions were analyzed. The effects of the gas–liquid ratio, overflow weir height, and absorption pressure on the absorption performance of each component were studied, and the influence of the acid component concentration in the feed gas on the removal efficiency of methyl mercaptan (CH3SH) was also discussed. It is found that the improved absorption model can better characterize the absorption performance and be conducive to the optimal design of the absorber column.