The “CO2 saturation effect” is a fallacy, based on a misunderstanding of heat transfer in the atmosphere. For details, see https://skepticalscience.com/saturated-co2-effect.htm
In brief, yes, the CO2 absorption in the earth’s atmosphere has been saturated for a long time. Only the upper atmosphere radiates heat into space (in the absorption wavelength band). In the lower atmosphere, heat is transferred upwards by convection.
Increase the CO2 percentage in the atmosphere and you raise the altitude of the convection/radiation transition. To maintain the same heat radiation, you need the same temperature, but at a higher altitude. Because of the atmospheric temperature lapse rate, that means you need a higher surface / sea level temperature.