Maser polarization is analyzed in the limit of overlapping Zeeman components (gv(B) much less than DELTAv(D), where gv(B) is the Zeeman splitting and DELTAnu(D) is the Doppler width). All the discrepancies among the conflicting conclusions of previous studies that identified maser polarizations in this limit are fully resolved. In the case of m-independent pumping, proper application of the eigenvalue technique of Goldreich, Keeley, & Kwan shows that the polarization eigenvectors are the same for saturated and unsaturated masers and are independent of spin for pure spin states, in agreement with the results of the first paper in this series. Stable eigenvectors correspond to the peak of the polarization mode distribution of self-amplified radiation at any degree of saturation. But the distribution average, the actually measured polarization, does not necessarily coincide with its peak. The mode distribution starts with a rectangular shape, because the seed radiation generated in spontaneous decays is unpolarized. and evolves toward a sharply peaked profile whose average, and not just its peak, coincides with the eigenvector solution because of the following two effects. First, interaction with the maser molecules induces rotation of the polarization vectors of individual modes, similar to Faraday rotation. The rotation rate is different for different modes, and the polarization eigenvectors correspond to stationary modes that do not rotate. Starting from unpolarized radiation generated by the source terms and containing an equal mix of all modes, all individual polarization vectors rotate into the stationary stable modes, resulting in a radiation field polarized according to the solution of the eigenvalue problem. As a result of this rotation the ensemble-averaged Stokes parameters reach the eigenvector solution when J greater than or similar to J(s), where J(s) is the angle-averaged intensity and J. is the saturation intensity, i.e., only after the maser saturates. This explains the results of numerical studies of the maser polarization problem presented in the literature. Second, and more important, maser growth is highly unstable during the unsaturated phase for any polarization configuration except for that of the eigenvector solution. The Stokes parameters of all other polarization structures include terms proportional to exp \aI\, where I is the intensity and a not-equal 0, and thus are highly unstable against arbitrarily small intensity perturbations. Such perturbations induce runaway divergence of the ensemble-averaged Stokes parameters away from their initial values, a divergence that stops only when the polarization settles into the appropriate eigenvector solution. The e-folding growth rate of the instabilities increases with J and reaches unity when J approximately J(s)/tau(s), where tau(s) is the optical depth of the maser when it saturates; pumping schemes of astronomical masers typically produce tau(s) approximately 12-17. Instabilities impose an upper bound on the intensity of radiation whose polarization differs from that of the eigenvector solution and are the dominant factor in narrowing the polarization mode distribution around its peak. Only radiation whose ensemble-averaged polarization corresponds to the eigenvector solution can grow to saturation and beyond. Furthermore, all polarization configurations are unstable for propagation at 0 < theta < theta0, where theta is measured from the magnetic axis and sin2 theta0 = 1/3. One eigenvector solution, corresponding to fully polarized radiation, is stable in this region during the unsaturated growth phase against perturbations that rotate the polarization at a fixed intensity, but not against intensity perturbations. As a result, stable buildup of maser radiation in a magnetic field with gv(B) much less than DELTAv(D) is possible only for theta greater-than-or-equal-to 0,; propagation directions too close to the field axis, corresponding to a fractional volume of approximately 0.09, are excluded. Propagation along the axis, theta = 0, is allowed, but the corresponding radiation is unpolarized.
