Well in my personal experience the quieter and less conspicuous I'm trying to be the more earth-shakingly loud everything I do seems to be and the more I draw attention to myself. This is especially true when I comes to having a house full of sleeping babies and toddlers…I try to tiptoe around the place and guaranteed I wake someone up…I carelessly walk around at night half awake, stub my toe, shout in pain and surprise, step on a plastic stegosaurus whose spines dig in real good for another angry grunt, open the fridge to get the pickles my half awake brain needed but open the door too violently and several other items fall out and crash to the floor…and nobody wakes up. Nobody notices. Not even the dog can be bothered to see what all the commotion is.
All of this is closely based on true events.
Moreover the number of times I've been surprised by an animal being close to me, not when I'm hunting but just out and about, doing my own thing, not taking care to be quiet, convinces me that it's not the end of the world if you're not perfectly silent and invisible.
Based on this "data", I say act natural and cough if you have to cough. Don't blow an eardrum suppressing a sneeze. Don't be so occupied with trying not to cough that you can't focus on your surroundings.
In fact, I say let er' buck and put some extra effort into hacking and wheezing full blast, really emphasize the "ah" before the choo, make a spectacle of yourself. The animals will either be awestruck and paralyzed with wonder at what the heck that noise is, or they'll conclude that whatever on earth is making that horrendous noise sure isn't trying to sneak up on them, nothing suspicious here, and it sounds like it's having a hard time so probably not a threat…
Your mileage may vary.