In this work, we study a Z3-symmetric extension of the Standard Model with three hyperchargeless SU (2)L fermion triplets responsible for neutrino mass generation via the Type-III seesaw mechanism together with a complex scalar singlet χ whose vacuum expectation value spontaneously breaks the Z3 symmetry. Radiative corrections induced by the Yukawa interactions between the SU (2)L fermion triplets and the complex scalar singlet χ generate a Coleman-Weinberg vacuum bias that lifts the degeneracy among the Z3 vacua, leading to the annihilation of unstable domain-walls. Consequently, the degeneracy among the Z3 vacua is lifted radiatively through the Coleman-Weinberg effective potential, generating a dynamical bias term that triggers the annihilation of unstable domain walls. We perform a numerical analysis consistent with current neutrino oscillation data and identify viable regions of parameter space accommodating the observed neutrino masses and leptonic mixing parameters. The observed baryon asymmetry of the Universe is generated through thermal leptogenesis via the out-of-equilibrium decay of the lightest fermion triplet for masses around O(10^9) GeV, consistent with the Type-III seesaw framework. Depending on the choice of model parameters, the predicted gravitational-wave spectrum can fall within the sensitivity reach of future space-based and ground-based gravitational-wave detectors. Our framework therefore establishes a correlation between neutrino mass generation, leptogenesis, radiative domain-wall instability, and gravitational-wave phenomenology.