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The stability and other physical properties of a class of regular black holes, quasiblack holes, and other electrically charged compact objects are investigated in the present work. The compact objects are obtained by solving the Einstein-Maxwell system of equations assuming spherical symmetry in a static spacetime. The spacetime is split in two regions by a spherical surface of coordinate radius $a$. The interior region contains a nonisotropic charged fluid with a de Sitter type equation of state, $p_r = -ρ_m$, $p_r$ and $ρ_m$ being respectively the radial pressure and the energy density of the fluid. The charge distribution is chosen as a well behaved power-law function. The exterior region is the electrovacuum Reissner-Nordström metric, which is joined to the interior metric through a spherical thin shell (a thin matter layer) placed at the radius $a$. The matter of the shell is assumed to be a perfect fluid satisfying a linear barotropic equation of state, ${\cal P}=ωσ$, with ${\cal P}$ and $σ$ being respectively the pressure and energy density of the shell, with $ω$ being a constant. The exact solutions obtained are analyzed in some detail by exploring the interesting regions of parameter space, complementing the analysis of previous works on similar models. This is the first important contribution of the present study. The stability of the solutions are then investigated considering perturbations around the equilibrium position of the shell. This is the second and the most important contribution of this work. We find that there are stable objects in relatively large regions of the parameter space. In particular, there are stable regular black holes for all values of the parameter $ω$ of interest. Other stable ultracompact objects as quasiblack holes, gravastars, and even overcharged stars are allowed in certain regions of the parameter space.