The dwarf spheroidal galaxies (dSphs) in the Milky Way are the primary targets in the indirect searches for particle dark matter. To set robust constraints on candidate of dark matter particles, understanding the dark halo structure of these systems is of substantial importance. In this paper, we first evaluate the astrophysical factors for dark matter annihilation and decay for 24 dSphs, with taking into account a non-spherical dark halo, using generalized axisymmetric mass models based on axisymmetric Jeans equations. First, from a fitting analysis of the most recent kinematic data available, our axisymmetric mass models are a much better fit than previous spherical ones, thus, our work should be the most realistic and reliable estimator for astrophysical factors. Secondly, we find that among analysed dSphs, the ultra-faint dwarf galaxies Triangulum II and Ursa Major II are the most promising but large uncertain targets for dark matter annihilation while the classical dSph Draco is the most robust and detectable target for dark matter decay. It is also found that the non-sphericity of luminous and dark components influences the estimate of astrophysical factors, even though these factors largely depend on the sample size, the prior range of parameters and the spatial extent of the dark halo. Moreover, owing to these effects, the constraints on the dark matter annihilation cross-section are more conservative than those of previous spherical works. These results are important for optimizing and designing dark matter searches in current and future multi-messenger observations by space and ground-based telescopes.