The random phase screens in anisotropic non-Kolmogorov turbulence are generated by the non-uniform sampling and power spectral inversion method, and the spatial optical modulator is adopted to simulate the drift characteristics and the changes of the orbital angular momentum of ring Airy Gaussian vortex beams in atmospheric turbulence. Numerical simulation and optical experiment results show that the drift values of ring Airy Gaussian vortex beams increase with the increasing beam attenuation coefficients, radius of the primary ring, outer scale of the atmospheric turbulence, and transmission distance, and decrease with the increasing turbulent anisotropic coefficients and topological charge of the beams. There exists a maximum drift value near the turbulent power-law value of 3.3. Moreover, comparing the interference fringe patterns of ring Airy Gaussian bortex beams before and after propagating through the atmospheric turbulence, we find that the smaller the topological charge is, the better the stability of topological charge after beam propagating in the turbulence is.