Want to move mother to a better nursing home on Medicaid, can I?

Once you find a good one there might be a wait list of several months. You might have to pay for the transport

I was told to go to the NH I wanted to move aunt to. They will take care of transfer. This will depend on your area as to what is available.

You have to stay in the same State, Medicaid does not transfer from State to State.

Current facility residents get "first dibs" on that facility's Medicaid beds. If you move your Mom out, she may have to go onto a waiting list at better places. It all depends on your location and the number of facilities there are (like if you're in an urban area there'll be many more options).

Yes, but it is not easy. You have to find a place you like and ask if they have open beds and will accept her.