[Archduke Tips]

Before you go replacing anything, first you should check if the problem is noise. Electrical noise can cause artifacts in audio and video. There is no easy way for you to check for noise (it is not usually audible) without at least an oscilloscope, but there are things you can try.

1.) Move speakers (especially big ones and bass) away from the cables.
2.) Unplug other electronics that are on the same circuit with the TV and XBOX. Especially look for things with motors or large vacuum tubes (such as refrigerators or old CRT type monitors). You can't just move them to another plug, they need to be on a physically different circuit (breaker).
3.) Move anything with a wireless transmitter away.
4.) If you use an RF-modulator for coax then get rid of it, they create tons of noise.

On analog signals noise usually shows up as snow, but HDMI uses a digital signal so it would do something unexpected when it has a bit error due to noise (like make jaggies maybe?). Try out what I said and let me know if it helps or not.