Bosonic Dirac materials are testbeds for dissipationless spin-based electronics. In the quasi two-dimensional honeycomb lattice of CrX₃ (X = Cl, Br, I), Dirac magnons have been predicted at the crossing of acoustical and optical spin waves, analogous to Dirac fermions in graphene. Here we show that, distinct from CrBr₃ and CrI₃, gapless Dirac magnons are present in bulk CrCl₃, with inelastic neutron scattering intensity at low temperatures approaching zero at the Dirac K point. Upon warming, magnon-magnon interactions induce strong renormalization and decreased lifetimes, with a ~25% softening of the upper magnon branch intensity from 5 to 50 K, though magnon features persist well above T_N. Moreover, on cooling below ~50 K, an anomalous increase in the a-axis lattice constant and a hardening of a ~26 meV phonon feature are observed, indicating magnetoelastic and spin-phonon coupling arising from an increase in the in-plane spin correlations that begins tens of Kelvin above T_N.