One of the main achievements in modern cosmology is the so-called `unified model', which successfully describes most classes of active galactic nuclei (AGN) within a single physical scheme. However, there is a particular class of radio-luminous AGN that presently cannot be explained within this framework -- the `low-excitation' radio AGN (LERAGN). Recently, a scenario has been put forward which predicts that LERAGN, and their regular `high-excitation' radio AGN (HERAGN) counterparts represent different (red sequence vs. green valley) phases of galaxy evolution. These different evolutionary states are also expected to be reflected in their host galaxy properties, in particular their cold gas content. To test this, here we present CO(1-0) observations toward a sample of 11 of these systems conducted with CARMA. Combining our observations with literature data, we derive molecular gas masses (or upper limits) for a complete, representative, sample of 21 z<0.1 radio AGN. Our results yield that HERAGN on average have a factor of ~7 higher gas masses than LERAGN. We also infer younger stellar ages, lower stellar, halo, and central supermassive black masses, as well as higher black hole accretion efficiencies in HERAGN relative to LERAGN. These findings support the idea that high- and low-excitation radio AGN form two physically distinct populations of galaxies that reflect different stages of massive galaxy build-up.