Quite some melee attacks and even some ranged ones do include a quick character movement animation - very obviously in some cases (e.g. the iconic backwards somersault on a Demon Hunters’ “Eye Beam”) and a lot more subtle in others (like slightly leaning into the shield and shifting the centre of gravity on a “Shield Slam”).
Actual screen shaking (outside a select few dungeon/raid-boss abilities) should however usually not happen, and if the effects that do occur are causing physical trouble like the previously mentioned headaches for anyone we’d very much love to hear about it!