The fact that your pain moves from one side to another would seem to me to indicate that it is not a kidney stone,UNLESS you have a stone(s) in each of your two kidneys. I've had many stones in both sides and I've never had the symtoms that you are describing.
I've had the same pain, it comes and goes and switches sides. I'm waiting on a ct to figure this out. I had an infection with low grade fever last month but the infection has gone but the pains haven't