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Exact chirped elliptic wave solutions are obtained within the framework of coupled cubic nonlinear Helmholtz equations in the presence of non-Kerr nonlinearity like self steepening and self frequency shift. It is shown that, for a particular combination of the self steepening and the self frequency shift parameters, the associated nontrivial phase gives rise to chirp reversal across the solitary wave profile. But a different combination of non-Kerr terms leads to chirping but no chirp reversal. The effect of nonparaxial parameter on physical quantities such as intensity, speed and pulse-width of the elliptic waves is studied too. It is found that the speed of the solitary wave can be tuned by altering the nonparaxial parameter. Stable propagation of these nonparaxial elliptic waves is achieved by an appropriate choice of parameters.