If you are getting very obvious vignetting (I.e. it looks like your image is framed by a blacked out circle) I am willing to bet you have Auto DX Crop off in your shooting menu.
Under image size in your shooting menu turn it on and it will automatically put your D600 in DX crop mode when a DX lens is on it. In the VF you will see a box around the image that will be captured (as it is smaller than the FX image area).
In DX crop mode your 11-16 shouldn't vignette that badly at all.
The Tokina 11-16 is designed for DX cameras and will produce vignetting on full frame FX. I have this lens and I do use it with my D600, but only at 16mm and stopped down a little to f/5.6. It's a terrific lens for DX and can produce good usable results on FX but only at 15 or 16mm settings.