I do not know the equipment Markel is using as his pre-amp, but I have used Audacity a bit.
I suggest that there is something very wrong with his set-up and the suggestion by Arcaxis is quite possible as there are two things that I have observed from the initial photos that Markel posted, these being:
1 In the photos of his fader positions, I notice that every fader is fully up (the ideal position being about 75% the way up), but even at full fader the Vu meter readings are quite low. This suggests that there is either not enough signal from the microphone to drive the mic-preamp unit correctly, or that the mic-preamp does not have enough internal gain for the microphone.
2. In the photos of the various recordings he did into Audacity, the first 3 (I think there were four takes) are way too low in level, the last one being a far better level (but even it is a bit low). HOWEVER, if you look closely at the wave in the last take, you will see that it is nowhere near peeking within Audacity, BUT the tops/bottoms of the waves are completely Square Waved (ie completely distorted), hence the sound he is complaining about.
The Square Waving in Audacity, when it is not overloading the signal being recorded by the Audacity program can in my opinion ONLY be caused by the signal leaving the mic-preamp being distorted before it leaves the mic-preamp.
Again I state that I do not know the mic-preamp he is using, but I will bet my 40+years as a professional recording engineer, that his problem lies within the "gain structure" that he has set-up. Whether this be caused by the USB circuitry somewhere drawing too much current from the computer, the mic-preamp not receiving enough voltage/current from the USB cable and thus causing the electronics within the mic-preamp to be struggling, the settings of the gain structure within the mic-preamp being wrong (is he possibly using some form of line input rather than a mic input --- could explain the high fader position and low Vu reading), are the audio mixer settings on his computer's sound card set completely wrong, etc, etc, etc.
Whatever the problem, I would definitely say that it is "gain structure" related somewhere between his mic and the input to Audacity and from my experience, the understanding/setting of this, is the main problem most people have in getting a good/correct sound, especially if they are new to the industry.
Hope this helps a bit and unfortunately I will be in the studio most of the week, so might not be able to get back for awhile if further help is required.
David