Small warning for a slight rambling below. I haven't fully explored how the twins' auras affect those with nontraditional souls, so this is also a bit of brainstorming on my end. This might change slightly in the future, but these are my initial thoughts as to how it will work.
You can think of Dream and Nightmare's auras as applying a status effect in a limited area. Anyone within this bubble of sphere has the effect applied to them, but the effect changes slightly depending on the permissions the twins have access to.
In Classic's case, the auras manifest as a smell.
For Error, they manifest as physical sensations- almost like prickles on his marrow. They are highly unpleasant to him, so he tends to avoid them. And yes- the Error's glitches are what results in the aura becoming physical.
Any being whose soul is missing or whose soul is corrupted (I'm thinking of Killer here) experiences the auras in a highly muted way. They are able to subconsiously recognize what emotions the auras would normally manifest, but due to their lack of typical emotions, the emotions are not 'forced' on them.
Ink and Killer experience the auras in a similar way. However, the more Ink is off his paints, the more muted the auras become.
This question actually made me laugh because Nightmare distinctly NEVER weakens his aura around anyone. He enjoys watching others squirm, regardless of their allegiance.
However, it is possible to get used to his aura to a degree. Ccino and the Gang have developed this slight 'immunity'/familiarity with Nightmare's aura, so they are not quite as debilitated as others when so close to Nightmare. When Nightmare first brought each member of the Gang into the castle, he pretty much left them to adjust on their own. The Gang actually enjoys going on missions sometimes because it usually gives them an excuse to get out of the castle's atmosphere.
Killer had an easier time than the others, something they hate him for.
In terms of Cross, his soul is corrupted in that it is currently half monster and half human, but it is still complete. Therefore, the only thing that changes for him is that Chara is not exempt from Nightmare's aura.