HD668B

If you're really on a budget ($20 USD is tricky), the Sennheiser HD206 is your best bet. However, if you can stretch your budget a little, I can heartily recommend the Audio Technica ATH-M20x. A reasonable in-between option is the Samson SR850 (rebadged Superlux HD 681 or Superlux HD 668B). It's got

Ok this is going to sound silly but to start with you need to buy a pair you'll probably hate but are cheap and wired. Buy a paid of Superlux the HD681 are more neutral, HD668B are Treble heavy and HD681EVO are base heavy you don't need all 3 one will do. This is because almost every damn headphone

If you combine wires output and input into one plug it will sound like shit. Cord cannot handle two way audio quality well. You need some Soundblaster G3 or something with separate USB audio connection for mic but idk if it works for Xbox. though. Haven't tried JT1 but pretty sure it should be bette