We investigate the long-term dynamics of spin textures prepared by cooling unmagnetized spinor gases of F=1 ^87Rb to quantum degeneracy, observing domain coarsening and a strong dependence of the equilibration dynamics on the quadratic Zeeman shift q. For small values of |q|, the textures arrive at a configuration independent of the initial spin-state composition, characterized by large length-scale spin domains, and the establishment of easy-axis (negative q) or easy-plane (positive q)

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... anisotropy. For larger |q|, equilibration is delayed as the spin-state composition of the degenerate spinor gas remains close to its initial value. These observations support the mean-field equilibrium phase diagram predicted for a ferromagnetic spinor Bose-Einstein condensate, but also illustrate that equilibration is achieved under a narrow range of experimental settings, making the F=1 ^87Rb gas more suitable for studies of nonequilibrium quantum dynamics.