We investigate the quark spin-independent and spin-dependent distributions in the nucleon within the framework of a covariant and confining Nambu-Jona-Lasinio (NJL) model. The nucleon bound state is obtained by solving the Faddeev equation in the quark–diquark approximation. In contrast to earlier works, we include not only the scalar and axial vector diquark channels, but now also the pseudoscalar and vector diquarks. The inclusion of these new diquark channels is crucial in maintaining chiral symmetry. Since the scalar and axial vector diquarks have positive parity, they can be in an $s$-wave state when combined with a quark to form a nucleon, whereas pseudoscalar and vector diquarks have negative parity and so must be in a $p$-wave state. By including these $p$-wave diquark correlations, we seek to understand how they affect the quark helicity distributions. We present our calculations for the unpolarized and polarized quark light-front momentum distribution functions and compare them to the available empirical fits. We find the inclusion of these new diquark channels improved the agreement of our model with the empirically-parametrized distribution functions. We obtain the nucleon axial coupling $g_A$ from our calculation as $g_A=1.42$ for our full model, while $g_A=1.51$ if we include scalar and axial vector diquarks only. We do not find improvement on the spin sum from adding the pseudoscalar and vector diquarks.