This paper presents algorithms for non-zero sum nonlinear constrained dynamic games with full information. Such problems emerge when multiple players with action constraints and differing objectives interact over time. They model a wide range of applications include economics, defense, and energy systems. We show how to exploit the temporal structure in projected gradient and Douglas-Rachford (DR) splitting methods. The resulting algorithms converge locally to open-loop Nash equilibria (OLNE) at linear rates. Furthermore, we extend a stagewise Newton method to find a local feedback policy around an OLNE. In the special case of linear dynamics and polyhedral constraints, we show that this local feedback controller is an approximate feedback Nash equilibrium.