  Many previous studies have explored models of dynamically interacting dark
energy scenarios instead of a cosmological constant and cold dark matter. This
work aims to offer a comprehensive investigation of the Yukawa-type interaction
between dark energy and dark matter where dark matter mass is also a function
of a dynamical scalar field value. Unlike previous work, instead of taking an
approximate solution, we numerically solve the Klein-Gordon equation that
governs the intricate interplay between these two components of the dark sector
along with the corresponding perturbations, using a suitable shooting algorithm
to determine precise initial conditions. We have conducted a thorough analysis
of this interaction using the Markov Chain Monte Carlo method, incorporating
diverse datasets such as Planck, BAO, Pantheon+, and SH$_0$ES. Our results, for
the first time, reveal a discernible detection of the coupling constant, which
encapsulates the strength of the dark sector interaction. We find non-zero
$\beta$ values of 0.0487, 0.0680, 0.0712, and 0.0822 ($68\%$ Confidence Limit)
corresponding to the inclusion of each dataset, respectively. Notably, the
addition of SH$_0$ES data led to a significant increase in the prominence of
$\beta$, suggesting the potential for new physics in the dark sector. This
study serves as a strategic road map for future endeavours with non-linear
cosmology, which will contribute to an enhanced understanding and refinement of
constraints on the dark sector interaction.
