The magneto-dielectric spectroscopy of La0.95Ca0.05CoO3 covering the crossover of spin states reveals the strong coupling of its spin and dipolar degrees of freedom. The signature of the spin-state transition at 30 K clearly manifests in the magnetization data at a 1 Tesla optimal field. Our Co L3,2-edge X-ray absorption spectrum on the doped specimen is consistent with its suppressed low-to-intermediate spin-state transition temperature at ∼30 K compared to ∼150 K, documented for pure LaCoO3. The dispersive activation step in the dielectric constant with the associated relaxation peak in imaginary permittivity characterize the allied influence of coexistent spin-states on the dielectric character. Dipolar relaxation in the low-spin regime below the transition temperature is partly segmental (Vogel-Fulcher-Tamman (VFT) kinetics) and features magnetic-field tunability, whereas in the low/intermediate-spin disordered state above ∼30 K, it is uncorrelated (Arrhenic kinetics) and almost impervious to the magnetic field H. Kinetics-switchover defines the dipolar-glass transition temperature Tg(H) (=27 K|0T), below which their magneto-thermally-activated cooperative relaxations freeze out by the VFT temperature T0(H) (=15 K|0T). An applied magnetic field facilitates thermal activation in toggling the low spins up into the intermediate states. Consequently, the downsized dipolar-glass segments in the low-spin state and the independent dipoles in the intermediate state exhibit accelerated dynamics. A critical 5 Tesla field collapses the entire relaxation kinetics into a single Arrhenic behaviour, signaling that the dipolar glass is completely devitrified under all higher fields. The magneto-electricity (ME) spanning sizeable thermo-spectral range registers diverse signatures here in kinetic, spectral, and field behaviors, in contrast to the static/perturbative ME observed close to the spin-ordering in typical multiferroics. Intrinsic magneto-dielectricity (50%) along with vanishing magneto-loss is obtained at (27 K/50 kHz)9T. The sub-linear deviant and field-hysteretic split seen in above 4 Tesla suggests the emergence of robust dipoles organized into nano-clusters, induced by the internally-generated high magneto-electric field. An elaborate ω-T multi-dispersions diagram maps the rich variety of phase/response patterns, revealing highly-interacting magnetic and electric moments in the system.