It is good to understand the basics of Pixel Aspect Ratio (PAR) and Display Aspect Ratio(DAR).
The Dreamcast Video outputs a DAR of 4:3, i.e. 640x480.
The "Widescreen hack" changes the PAR and expects the TV or monitor will do the work of stretching the 4:3 image to 16:9, at 480p that would be ~853x480 display.
Think of it as writing to a 853x480 framebuffer, then scaling the 853x480 framebuffer down to 640x480 for video output by the PVR.
Then, the TV will stretch the 640x480 image back to 853x480.
Notice that is why we actually see more of the world to the left and right of the characters in the screens you posted there.
And BlueCrab is right, in 3D vertex submission, the only difference is the perspective matrix.
That is how I enable the option in my game engine:
viewtopic.php?f=29&t=102506&p=1047897#p1047897
But my guess is you do not transform your 2D vertices, submitting your sprites in the range of (0->640 , 0->480)
Basically, you need to scale your vertices' x component so that when stretched out by your TV into WideScreen, your desired pixel ratio is preserved.
One way to do this would be to re-organize your sprites for a 853x480 display.
Then, when submitting the vertices, do a simple transform on the x component like:
To scale the vertices from 0->853 back to 0->640.