E.A.
We've got an 8-week old who was doing the same occasional sneezing and some dry coughing. The lack of rain has made it so dry around here that his nasal passages and the mucus in them was drying up and kind of sticking. It should be flowing smoothly and keeping the little nose and throat moist.
We loaded up on distilled water and have been running a cool mist vaporizer over his bassinette at night. It did the trick and he's been breathing perfectly ever since. Hope it's just that simple for you, too.