The biaxiality of the steady state solutions and their stability to out-of-plane disturbances in shear flows of spatially homogeneous liquid crystal polymers using two approximate models, BMAB-Doi and BMAB-HL1, which are derived from the kinetic theory developed by Bhave et al. (1993) (BMAB) using the Doi and the first Hinch-Leal closure approximation, respectively, are studied. By casting the models in a novel biaxial representation of the orientation tensor with two built-in order parameters and a triad of directors, we show explicitly that the steady states of the BMAB models exhibit biaxial symmetry except for some uniaxial degeneracy at isolated Peclet numbers and polymer concentration values. Moreover, we obtain all the steady states in which two directors are confined to the shearing plane and analyze their stability with respect to both in-plane and out-of-plane disturbances. We find that (1) flow-aligning family is the unique stable solution family in the BMAB-Doi model, where two order parameters are of opposite signs; (2) the flow-aligning family in the BMAB-HL1 model is stable only in a finite range of polymer concentration 0 < N less than or equal to 10, the log-rolling family is born unstable and attains stability through an instability to stability transition at a sufficiently high polymer concentration value, N > 10, which grows with respect to the Peclet number; (3) The loss-of-stability in the flow-aligning family at N = 10 is caused by a one-dimensional director rotational instability pertinent to the existence of the maximum allowable degree of orientation with respect to the how-aligning major director, 5/6, and is coincident with the change-of-sign behavior of the first normal stress difference and the smaller order parameter as well.