We study the qualitative and numerical behavior of a thermodiffusion Timoshenko beam with Kelvin–Voigt damping. The model, recently derived by Aouadi $ \rm et al. $ (2019), describes the behavior of a Timoshenko beam with thermodiffusion effects under the classical Fourier's and Fick's laws for heat and mass diffusion transmission, respectively. With the help of the semigroup theory of linear operators, we prove the well posedness of the considered problem. Without assuming the well-known condition of equal wave speeds, it is shown that the associated $ C_0 $-semigroup to the fully viscoelastic system is analytic. Exponential stability and impossibility of localization of the solutions in time are immediate consequences. Two partial viscoelastic systems are shown to be polynomially or exponentially stable. The approach based on the Gearhart–Herbst–Prüss–Huang theorem is used to show the lack of exponential stability, whereas the optimal polynomial decay rate can be shown using the Borichev & Tomilov's Theorem. Finally, we propose a spectral discretization method to prove that the associated discrete energy decays exponentially or polynomially otherwise.