We demonstrate via exact diagonalization that AA-stacked transition metal dichalcogenide homobilayers host fractional quantum anomalous Hall (FQAH) states with fractionally quantized Hall conductance at fractional fillings n=13,23 and zero magnetic field. While both states are most robust at angles close to 2-degree, the n=13 state gives way to a charge density wave with increasing twist angle whereas the n=23 state survives across a much broader range of twist angles. We show that the competition between FQAH states and charge density wave or metallic phases is primarily controlled by the wave functions and dispersion of the underlying Chern band, respectively. Additionally, Ising ferromagnetism is found across a broad range of fillings where the system is insulating or metallic alike. The spin gap is enhanced at filling fractions where integer and fractional quantum anomalous Hall states are formed.