Understanding how galaxies cease to form stars represents an outstanding challenge for galaxy evolution theories. This process of "star formation quenching" has been related to various causes, including active galactic nuclei activity, the influence of large-scale dynamics, and the environment in which galaxies live. In this paper, we present the first results from a follow-up of CALIFA survey galaxies with observations of molecular gas obtained with the APEX telescope. Together with the EDGE-CARMA observations, we collected 12CO observations that cover approximately one effective radius in 472 CALIFA galaxies. We observe that the deficit of galaxy star formation with respect to the star formation main sequence (SFMS) increases with the absence of molecular gas and with a reduced efficiency of conversion of molecular gas into stars, which is in line with the results of other integrated studies. However, by dividing the sample into galaxies dominated by star formation and galaxies quenched in their centres (as indicated by the average value of the Hα equivalent width), we find that this deficit increases sharply once a certain level of gas consumption is reached, indicating that different mechanisms drive separation from the SFMS in star-forming and quenched galaxies. Our results indicate that differences in the amount of molecular gas at a fixed stellar mass are the primary drivers for the dispersion in the SFMS, and the most likely explanation for the start of star formation quenching. However, once a galaxy is quenched, changes in star formation efficiency drive how much a retired galaxy differs in its star formation rate from star-forming ones of similar masses. In other words, once a paucity of molecular gas has significantly reduced star formation, changes in the star formation efficiency are what drives a galaxy deeper into the red cloud, hence retiring it.