We report new, sensitive observations of two z ∼ 3–3.5 far-infrared-luminous radio galaxies, 6C 1909+72 and B3 J2330+3927, in the 12CO J = 1−0 transition with the Karl Jansky Very Large Array and at 100–500 m using Herschel, alongside new and archival 12CO J = 4−3 observations from the Plateau de Bure Interferometer. We introduce a new colour–colour diagnostic plot to constrain the redshifts of several distant, dusty galaxies in our target fields. A bright SMG near 6C 1909+72 likely shares the same node or filament as the signpost active galactic nuclei (AGN), but it is not detected in 12CO despite ∼20 000 km s−1 of velocity coverage. Also in the 6C 1909+72 field, a large, red dust feature spanning ≈500 kpc is aligned with the radio jet. We suggest several processes by which metal-rich material may have been transported, favouring a collimated outflow reminiscent of the jet-oriented metal enrichment seen in local cluster environments. Our interferometric imaging reveals a gas-rich companion to B3 J2330+3927; indeed, all bar one of the eight z ≳ 2 radio galaxies (or companions) detected in 12CO provide some evidence that starburst activity in radio-loud AGN at high redshift is driven by the interaction of two or more gas-rich systems in which a significant mass of stars has already formed, rather than via steady accretion of cold gas from the cosmic web. We find that the 12CO brightness temperature ratios in radio-loud AGN host galaxies are significantly higher than those seen in similarly intense starbursts where AGN activity is less pronounced. Our most extreme example, where , provides evidence that significant energy is being deposited rapidly into the molecular gas via X-rays and/or mechanical (‘quasar-mode’) feedback from the AGN, leading to a high degree of turbulence globally and a low optical depth in 12CO – feedback that may lead to the cessation of star formation on a time-scale commensurate with that of the jet activity, ≲10 Myr.