The flavor evolution of neutrinos in a dense environment is dependent on neutrino-neutrino interactions. Traditionally approached with quantum kinetic equations, recent works have used a quantum many-body approach using a forward-only Hamiltonian. Continuing a previous work (Cirigliano et al., Phys. Rev. D (2024) 110:123028), we study neutrino flavor evolution of a two-neutrino system using the Full neutrino-neutrino Hamiltonian. We extend the previous work using plane waves to Gaussian wave packet states. We discuss the time evolution of one-body flavor and kinetic observables, and the one-body entanglement entropy. We observe non-local interactions with the forward-only Hamiltonian, in contrast to local time evolution with the full Hamiltonian. The full Hamiltonian also demonstrates significantly faster time evolution. We demonstrate consistency with the previous studies in the case of plane waves. We also observe the interaction progressing in coordinate space, which confirms our results on locality, but also displays unexpected asymmetric behavior.