We investigate nonlinear effects on the dynamics of entanglement and other quantum observables in a system of two harmonic modes coupled through angular momentum. The nonlinearity arises from a Kerr-type anharmonic term in each mode. The emergence and evolution of entanglement, non-Gaussianity, photon number, photon antibunching and squeezing are examined for different initial coherent product states and couplings, through exact diagonalization in a truncated basis. It is shown that the anharmonic terms, even if weak, can lead to very significant effects for such initial states, considerably enhancing and stabilizing entanglement and leading to a non negligible non-Gaussianity of the evolved states. They also affect other observables, stabilizing the dynamics after an initial transient regime, for not too small initial average populations of each mode. Analytic short-time approximate expressions are also provided.