The DM particles would indeed be on random/chaotic orbits, unlike visible matter which can shed kinetic energy via EM interaction (collisions). But DM density near a mass concentration would still be higher than far away from anything massive.
Normal stars are in hydrostatic equilibrium, a density where the inward force exerted by gravity and the outward force exerted by the pressure of the hot plasma are balanced. In a dark star the situation would be similar, except the heat would be generated by DM annihilation rather than fusion (the heat from annihilation would keep the star too "puffy" to reach the core pressure and temperature required for fusion.
Normal stars are in hydrostatic equilibrium, a density where the inward force exerted by gravity and the outward force exerted by the pressure of the hot plasma are balanced. In a dark star the situation would be similar, except the heat would be generated by DM annihilation rather than fusion (the heat from annihilation would keep the star too "puffy" to reach the core pressure and temperature required for fusion.