The effects of different vortex breakdown states on the evaporation process characterizing air-acetone vapor swirling jets laden with liquid acetone droplets in the dilute regime are discussed based on results provided by direct numerical simulations. Adopting the point-droplet approximation, the carrier phase is solved using an Eulerian framework, whereas a Lagrangian tracking of the dispersed phase is used. Three test cases are investigated: one with fully-turbulent pipe inflow conditions and two with a laminar Maxworthy velocity profile at different swirl rates. Consequently, turbulent, bubble-type and regular conical vortex breakdown states are established. Following phenomenological and statistical analyses of both phases, a significant enhancement of the overall droplet evaporation process due to the onset of the conical vortex breakdown is observed due to the strongest centrifugal forces driving the entire liquid drops towards the low-saturation mixing layer of the jet. The effects of droplet inertia on evaporation are isolated through an additional set of simulations where liquid droplets are treated as Lagrangian tracers. While it is found that inertial effects contribute to enhanced vaporization near the mixing layer under bubble vortex breakdown conditions, droplet inertia plays a secondary role under both turbulent and conical vortex breakdown due to intense turbulent mixing and high centrifugal forces, respectively.