Context. The chemical and physical evolution of starless and pre-stellar cores are of paramount importance to understanding the process of star formation. The Taurus Molecular Cloud cores TMC 1-C and TMC 1-CP share similar initial conditions and provide an excellent opportunity to understand the evolution of the pre-stellar core phase. Aims. We investigated the evolutionary stage of starless cores based on observations towards the prototypical dark cores TMC 1-C and TMC 1-CP. Methods. We mapped the prototypical dark cores TMC 1-C and TMC 1-CP in the CS 3 → 2, C34S 3 → 2, 13CS 2 → 1, DCN 1 → 0, DCN 2 → 1, DNC 1 → 0, DNC 2 → 1, DN13C 1 → 0, DN13C 2 → 1, N2H+ 1 → 0, and N2D+ 1 → 0 transitions. We performed a multi-transitional study of CS and its isotopologs, DCN, and DNC lines to characterize the physical and chemical properties of these cores. We studied their chemistry using the state-of-the-art gas-grain chemical code NAUTILUS and pseudo time-dependent models to determine their evolutionary stage. Results. The central nH volume density, the N2H+ column density, and the abundances of deuterated species are higher in TMC 1-C than in TMC 1-CP, yielding a higher N2H+ deuterium fraction in TMC 1-C, thus indicating a later evolutionary stage for TMC 1-C. The chemical modeling with pseudo time-dependent models and their radiative transfer are in agreement with this statement, allowing us to estimate a collapse timescale of ~1 Myr for TMC 1-C. Models with a younger collapse scenario or a collapse slowed down by a magnetic support are found to more closely reproduce the observations towards TMC 1-CP. Conclusions. Observational diagnostics seem to indicate that TMC 1-C is in a later evolutionary stage than TMC 1-CP, with a chemical age ~1 Myr. TMC 1-C shows signs of being an evolved core at the onset of star formation, while TMC 1-CP appears to be in an earlier evolutionary stage due to a more recent formation or, alternatively, a collapse slowed down by a magnetic support.