I ended up just going back to the 3.5mm jack for audio in my cabinet. Based on the linked article, I just made sure I had the following settings in my config.txt:
disable_audio_dither=1
audio_pwm_mode=2
For some reason my pwm_mode was set to 0 in my config and changing that to 2 resolved all my issues with the 3.5mm jack.
Unfortunately, I was never able to get the crackle / hiss / static noise issue resolved when sending audio out HDMI to my HDMI to VGA adapter.