We investigate exact soliton solutions with nonlinear chirp for the coupled nonlinear Schrödinger equations with cubic-quintic nonlinearity, self-steepening, self-frequency shift and four-wave mixing. The model governs the femtosecond pulse propagation in birefringent fibers. We introduce a new ansatz to obtain the nonlinear chirp associated with the propagating soliton pulses. New chirped soliton pair solutions with non-trivial chirping are found for the coupled nonlinear equations, illustrating the potentially rich set of solitonic pulse solutions of the model with higher-order effects. The solutions comprise two types of bright-W-shaped and bright-bright soliton pairs as well as kink and anti-kink pulses. Interestingly, the bright wave in the bright-W shaped soliton pairs possesses a platform underneath it, originating from the self-steepening and self-frequency shift effects. The corresponding chirp associated with each of these optical soliton pairs is also determined. It is shown that the nonlinear chirp is related to the pair intensity and determined by self-frequency shift and pause self-steepening. Parametric conditions for the existence and uniqueness of chirped solutions are given.