Dwarf spheroidal galaxies (dSphs) are prime laboratories for studying dark matter (DM) and the black hole demographics in the low-mass regime. Although these systems are also often significantly flattened, most studies have relied on spherical models, which may affect dynamical inferences. In this work we introduce the first multi-component axisymmetric dynamical models of dSphs based on analytic distribution functions and apply them to the Milky Way dSphs Draco and Ursa Minor. The stellar distribution is described by chemo-dynamically distinct axisymmetric populations tracing a common spherical gravitational potential generated by a dominant DM halo and a central intermediate-mass black hole (IMBH). The models were fitted to discrete stellar data from an astrometrically selected sample of stellar coordinates based on Gaia and one of the two different spectroscopic datasets providing line-of-sight velocities and metallicities. This approach allowed us to test the robustness of our inferences against the use of different kinematic samples. We compared the inferred DM properties under different modelling assumptions, including flattened one-component and spherical two-component models. We find that both galaxies are better described by two stellar populations with distinct chemo-dynamical properties: a metal-rich component that is kinematically colder and more centrally concentrated and a more extended metal-poor component characterised by hotter kinematics. We detected weak signatures of rotation in both galaxies, which are dynamically unimportant and were ignored in the models. We measured a cuspy DM density profile in Draco (γ = 0.98−0.26+0.28), while Ursa Minor is consistent with a more cored distribution (γ = 0.37−0.24+0.31). We find that the DM profile of Draco remains stable across all models and datasets, making its DM halo the most robustly determined among Local Group dSphs and extremely relevant in the context of indirect DM searches. We also show that modelling intrinsically flattened systems with spherical models can bias the DM inner slope towards cuspier values, while we found no degeneracy between inner halo density and galaxy inclination. Finally, we found no evidence of IMBHs, and we placed upper limits on their masses, log10 MBH[ M⊙] < 5.2 for Draco and <3.33 for Ursa Minor (95% confidence), with the latter providing a particularly stringent constraint.