Best answer as chosen by user TheMegaStorm: The bit about
The bit about how the plug doesn't fit all the way into the jack seems telling. The first thing I'd do personally, is find some cheap pair of headphones and plug them into the TV to see if the symptoms persist across multiple headsets. If they do, take the TV back and exchange it for another one because I'd say you got a defective one. In the highly unlikely event that the second set is defective in exactly the same way, the TV manufacturer is probably using some non-standard design for that jack. You can try checking the manual to see if it has anything to say about it, but you may just have to contact the manufacturer who will want to sell you some kind of accessory for that. However, that is pretty unlikely IMO.