We address polarization instability in a freely decaying polariton condensate created by 2-ps-long linearly polarized laser pulses in the upper sublevel of the lower-polariton (LP) branch in a GaAs-based microcavity with reduced symmetry. The generated linearly polarized condensate is found to lose its stability at excitation densities above the threshold value: it passes into the regime of inner Josephson oscillations with strongly oscillating circular and diagonal linear polarization degrees, as well as monotonically decreased oscillations in linear polarization accompanied by a gradual increase in the condensate of the low-sublevel component. These phenomena occur with a relatively small decrease in the total polarization and spatial coherence of the spinor condensate. At high LP densities, the LP-LP interaction leads to the nonlinear Josephson effect. All effects are found to be well reproduced by the model based on spinor Gross-Pitaevskii equations. The cause of the instability was clarified by considering a simplified model of the spinor 0D oscillator: it is spin anisotropy of the LP-LP interaction. The threshold density was shown to increase with decreasing difference Δα of the constants of the interparticle interaction of LPs with identical and opposite spins as δl/Δα, where δl is the LP level splitting. The reduction in the linear polarization is connected with the fact that the LPs escaping from the condensate with oscillating circular polarization carry off more energy than from the linearly polarized one.