MAX7221 would be a better chip for this situation because it's supposed to have lower EMI.
What makes you say that?
Any high speed digital signals generate signals that can be picked up on audio circuits, it is quite common.
You need to make sure your analogue ground and your digital ground are kept separate and only joined at one point. Keep digital lines as far away as possible from analogue lines. Make sure the digital power supply is separate or at least decoupled from the analogue supply.
You can also limit the bandwidth of the audio circuits so they don't pick up as much.
Yes tiding up that rats nest will help but without correct layout it will not be enough.