Welcome, I spend most of my life in a state of confusion
Unless Tamron makes two versions of the 200-400 it is a full frame lens. The specs indicate a 12 degree angle of view at 400mm which is correct for full frame. It is 8 degrees at 400mm for DX. Therefore as you stated the outer edges of the image don't fall on the sensor. When you look at MTF charts for full frame lenses they are graphed from 0 to 22mm on the x axis. That corresponds to the diagonal distance from center of the frame. The drastic fall of in sharpness typically occurs in the outer third from 15-22mm range. Nikon DX sensors only use the image from 0-14.5(nominally). Of course one can always find exceptions to general statements like this but statistically this is true more often than not.
A general statement can't be made that you will get less vignetting with an FX lens vs a DX lens. It is accurate to say that for a given FX lens you will get less vingette effect on a DX body than the same lens on an FX body.
For these reasons as previously stated, it is possible to get good sharpness and little vignette with less than top quality FX lenses used on DX bodies. Unfortunately poor contrast, poor color rendition, and high chormatic abberation are also typically characteristic of lower quality lenses and will be the same regardless of format.
Two examples of this are the Nikon 300mm f4 and the Sigma 100-300mm f4 both of which are tack sharp on DX bodies even when used with 1.4x extenders.