A chord is a combination of musical notes, OP. So it’s vocal cords, not chords! They’re two thin strips, or cords. at your laryngeal inlet, which move to alter the air flow and therefore the sound quality of your speech. They can close together completely to try and stop you inhaling toast crumbs, or go into spasm, causing a croupy crowing noise when you breathe.
It’s racking cough, not wracking. The rack was a torture device in mediaeval times, and being racked was incredibly painful, often dislocating joints. A racking cough is one so severe it feels like you’re being pulled apart. Wrack is a form of seaweed, eg bladder wrack.