We breathe by moving our diaphragm down, which creates a partial vacuum in our lungs (air pressure inside is lower than pressure outside the body). Instead of the outside air crushing the chest (which it will, a bit, if you try to inhale with your airway closed), the open airway allows air to rush in (it's pushed in by the surrounding air expanding into the airway).
Given that, it's hopefully easy to see why a larger opening to the airway allows for more air to enter, more quickly. The strength of the muscles moving the diaphragm is obviously a factor, too, which you might want to "tweak" in your story if you are trying to maximize air intake. But for the purposes of this answer, we can assume it's constant.
Inhaling through the nose means that difference in pressure has to be resolved by piping air through two very narrow little nostrils. The trachea isn't exactly roomy at 15-20cm inside diameter, so you don't exactly have to have your mouth wide open to maximize breathing, but it's going to be a lot better than nose breathing.
Solving the dryness problem
The humidity and temperature of the air coming in makes a big difference. (A colder ambient temperature really reduces the relative humidity when the air is heated in the airway.) Higher airflow means you're going to need more temperature and humidity, either within the airway, or perhaps even outside the body in a heated humidifier. Anyone who has ever slept with a CPAP machine, especially in a colder climate, knows this very well.
Other ways to get more oxygen
You may have already considered this as well, but humans breathe in 20% oxygen and exhale 15% oxygen, so we're not especially good at it. If your geneticists can improve that number by 5%, you double the available oxygen in each breath, over and above anything I've already discussed. Adding supplemental oxygen helps in a similar fashion.