Cold cores are an early step of star formation, characterized by densities > 10$^4$ cm$^{-3}$, low temperatures (< 15 K), and very low external UV radiation. We investigate the physico-chemical processes at play to tracing the origin of molecules that are predominantly formed via reactions on dust grain surfaces. We observed the cold core LDN 429-C with the NOEMA interferometer and the IRAM 30m single dish telescope in order to obtain the gas-phase abundances of key species, including CO and CH$_3$OH. Comparing the observed gas phase of methanol to its solid phase previously observed with Spitzer allows us to put quantitative constraints on the efficiency of the non-thermal desorption of this species. With physical parameters determined from available Herschel data, we computed abundance maps of 11 detected molecules with a non-local thermal equilibrium radiative transfer model. These observations allowed us to probe the molecular abundances as a function of density and visual extinction, with the variation in temperature being restrained between 12 and 18 K. We then compared the observed abundances to the predictions of the Nautilus astrochemical model. We find that all molecules have lower abundances at high densities and visual extinctions with respect to lower density regions, except for methanol. Comparing these observations with a grid of chemical models based on the local physical conditions, we were able to reproduce these observations, allowing only the parameter time to vary. Comparing the observed gas-phase abundance of methanol with previous measurements of the methanol ice, we estimate a non-thermal desorption efficiency between 0.002% and 0.09%, increasing with density. The apparent increase in the desorption efficiency cannot be reproduced by our model unless the yield of cosmic-ray sputtering is altered due to the ice composition varying as a function of density.Read less <