We present an analysis of the $e^{+}{e}^{-}\to {\gamma }^{⁎}\to J/\psi \pi \pi (K\overline{K})$ and $e^{+}{e}^{-}\to {\gamma }^{⁎}\to h_c\pi \pi$ processes employing the recently proposed Dalitz-plot decomposition approach, which is based on the helicity formalism for three-body decays. For the above reactions, we validate the factorization of the overall rotation for all decay chains and spin alignments, along with the crossing symmetry between final states, using a Lagrangian-based toy model. For the model-dependent factors that describe the subchannel dynamics, we employ the dispersive treatment of the $\pi \pi (K\overline{K})$ final state interaction, which accurately reproduces pole positions and couplings of the $f_0(500)$ and $f_0(980)$ resonances. The constructed amplitudes serve as an essential framework to further constrain the properties of the charged exotic states $Z_c(3900)$ and $Z_c(4020)$, produced in these reactions.