I went from d5100 to d610. Full frame and within your budget. Totally no regrets and loving it.
Downside is most people on DX already accumulated too many DX lenses and find it too painful to move to FX. Yet they don't want to buy FX lenses and use on DX because of the crop factor. Not like me, I stayed with FX lenses thru 7 years on DX before moving to FX, so I had no pain on my side.
Most friends that I tempted them to move from DX to FX felt the same, but they definitely had no regrets.
Made at least 5 friends jump to FX on D600/D610/D3S. They're loving it.
To jump from 3100 to 5100 to 7100 doesn't really make much difference as the sensor/AI is still better on FX.