Nasal pillows do "apply" the air pressure more directly into your nostrils than a nasal cushion or a full face mask. You may need a little more humidity if this is a recent problem. AND some Ayr Saline Nasal Gel or even Ayr Saline Nasal Spray tho I personally feel the Gel is better in this type of situation. NEVER use petroleum based products in or near the nostrils. It is detrimental to your health.
Also it could be that your headgear is not quite centered on your head and pulling your nasal pillows slightly more to one side than the other, or the hose is tugging your nasal pillows slight more to one side than the other if it is just one nostril and the pillow itself is rubbing the interior of your nose. That was a problem I had w/the original Swift. And you are right about the air "burning" .... it felt like someone pouring salt on a raw wound!