Assume that the motion of ions in plasma along one direction is governed by the 1D Navier-Stokes-Poisson equations with the Boltzmann relation for the electrons. For this model, we establish the existence of small-amplitude shock waves via center manifold theorem, and further prove its dynamical stability under suitable smooth perturbations in terms of the classical energy method. It is also justified that the propagation of shock profiles is dominated by the KdV-Burgers equation in a suitable asymptotic regime where the viscosity effect is stronger enough than the one of dispersion described by the Debye length.