The effect of the sign of the dispersion parameter on nonlinear pulse propagation in a birefringent optical fiber is analyzed via numerical analysis. First, the concept of effective cross-dispersion is introduced to demonstrate the simultaneous existence of positive dispersions of opposite polarizations within the same waveguide, leading to the generation of a dispersive wave and supercontinuum. Then, a general set of equations is deduced to calculate the dispersive waves corresponding to x and y polarizations considering a birefringence-dependent wavelength. Using this, it is established that solitons and dispersive waves can be induced corresponding to each polarization even if the dispersions are normal. Finally, we demonstrate that spectral broadening can also be induced corresponding to one polarization despite the existence of positive dispersion, i.e., in the complete absence of solitons for the entire duration of pulse propagation.