Differential cross sections have been extracted from exclusive and kinematically complete high-statistics measurements of quasifree polarized $\vec{n}p$ scattering performed in the energy region of the $d^{*}\left(2380\right)$ dibaryon resonance covering the range of beam energies $T_n=0.98-1.29$ GeV ($\sqrt{s}=2.32-2.44$ GeV). The experiment was carried out with the WASA-at-COSY setup having a polarized deuteron beam impinged on the hydrogen pellet target and utilizing the quasifree process $dp\to np+p_{\mathrm{spectator}}$. In this way the $np$ differential cross section $\sigma \left(\Theta \right)$ was measured over a large angular range. The obtained angular distributions complement the corresponding analyzing power $A_y\left(\Theta \right)$ measurements published previously. A SAID partial-wave analysis incorporating the new data strengthens the finding of a resonance pole in the coupled ${}^{3}D_3\text{−}{}^{3}G_3$ waves.