How abruptly does the Doppler shift change sign?

The instantaneous change occurs when you consider the Doppler shift in only one dimension. In three dimensions you can consider the correction when the velocity vector and the separation vector are not parallel. Usually such corrections go like $\cos\theta$, where $\theta$ is the angle between the two vectors, but more complicated things are possible.

Years ago I sat down and computed the speeds for which acoustic Doppler shifts correspond to musical intervals. That gave me the superpower of being able to stand on a sidewalk, listen to the WEEE-ooom as a car drove past, and say to myself “a major third? They're speeding!” But because of the $\cos\theta$ dependence, the trick gets harder as you get further from the road.

The object cannot occupy your same place as he passes you, so let us assume that the trajectory is a straight line that passes next to you. As the object approaches, the component of the velocity in your direction diminishes, to the point of being zero when the object is next to you. Thus the doppler effect will change continuously, from blue to zero to red shifted.

You can pretty well observe acoustic Doppler shift just by listening to moving objects emitting some more or less constant-frequency (i.e. tone) sound.

Motor cars (or even better, motorbikes) are pretty good for Doppler observations.

People are pretty much used to approach-passby-goaway sound pattern these objects make.

If you listen from a sidewalk, you hear a quick change to lower tone as a car passes by. The more distance from the road, the slower the change.

There is really no gross difference in EM Doppler effect.