  We place observational constraints on a dark energy (DE) model in which a
quintessence scalar field $\phi$ is coupled to dark matter (DM) through
momentum and energy exchanges.The momentum transfer is weighed by an
interaction between the field derivative and DM four velocity with a coupling
constant $\beta$, whereas the energy exchange is characterized by an
exponential scalar-field coupling to the DM density with a coupling constant
$Q$. A positive coupling $\beta$ leads to the suppression for the growth of DM
density perturbations at low redshifts, whose property offers a possibility for
resolving the $\sigma_8$ tension problem. A negative coupling $Q$ gives rise to
a $\phi$-matter-dominated epoch, whose presence can reduce the sound horizon
around the Cosmic Microwave Background (CMB) decoupling epoch. Using the data
of Planck 2018, 12-th Sloan Digital Sky Survey, Phantheon supernovae samples,
and 1-year dark energy survey, we find that the two couplings are constrained
to be $\beta=0.332^{+1.246}_{-0.237}$ and $Q =-0.0312^{+0.0312}_{-0.0085}$ at
68\,\% confidence level (CL). Thus, there is an interesting observational
signature of the momentum exchange ($\beta \neq 0$) between DE and DM, with a
peak of the probability distribution of the energy transfer coupling at $Q<0$.
