Your testing has already isolated this properly: two headsets, both fine elsewhere, both crackling in the controller, with the fault following cable movement. That is the controller's 3.5mm jack and nothing else.
There are two causes and they need different fixes. The first is contamination. A 3.5mm socket is an open hole on the bottom of a controller that lives on sofas and in bags, and it collects lint and dust that works its way onto the contact rings. A partially contaminated contact gives you crackling and intermittent dropouts that change as the plug moves, which matches your description precisely. This is the more common of the two and it costs nothing to address.
The second is cracked solder joints at the jack's mounting points, the same failure mode as a worn charging port. Every time the headset cable is tugged or levered, the joints take the strain. Once cracked, holding the plug at an angle closes the crack and it works, which is the other half of your description. So both causes can produce the symptom and you cannot tell them apart from outside.
Start with cleaning, properly. Compressed air into the socket at an angle, then a dry cotton bud twisted gently inside, or a contact cleaner designed for audio jacks sprayed on the plug rather than into the socket. Then insert and remove the plug twenty or thirty times, which scrubs the contacts, and re-test. A meaningful share of crackling jacks are fully resolved by this.
If it persists, the joints are cracked and that is a soldering repair. The jack is a cheap part and it is a routine job for a phone repair shop. It is not board-level in the difficult sense, as the jack is a discrete through-board component rather than something fine-pitch.
One practical habit either way: unplug by the plug, never by pulling the cable, and avoid letting the controller hang from the headset cable. That is what cracks these joints in the first place.
