In this paper, we consider a distributed design for generalized Nash equilibrium seeking of a game with both coupled and local constraints. We first propose a distributed continuous-time algorithm to approximate a generalized Nash equilibrium based on singular perturbation, where players take actions using their local constraint data and communicate through a weight-balanced network. Compared with exact-variational equilibrium (VE) dynamics, the proposed algorithm reduces the state dimension and the number of multiplier-related scalars transmitted per communication round. Then we show the existence of the approximate variational equilibrium, and moreover, provide an upper approximation bound, which depends on a tunable parameter given in the algorithm. Also, we prove that the approximate equilibrium approaches the exact one as the parameter tends to zero, and present the convergence analysis with the help of Lyapunov functions and variational inequalities. Experimental validations are provided to show the effectiveness of our approach.