@gtl- you could see a Neurologist to navigate the nerve sensitization with you. They would most likely treat this as a migraine type situation. They see a lot of patients that originally had “sinus headache” complaints that actually were really migraine type headaches. Migraines refer to the nose as congestion. They are also referred many patients after sinus surgeries.

-the cotton ball relief, sounds like it is slowing, redirecting, warming air, before hitting the sensible nerve area.
The same way someone wears ear muffs to block the cold air from ear irritation and or exits a building into the cold and puts a scarf over their nose for protection. Both of these responses are to slow/adjust the nerve sensitivity to the external cold air, until the body adjusts.
-Nerve irritation in the nose can cause a sensation of head fullness. Kind of like the nerve that gets activated when cold drinks are taken into quickly, the “ice cream” headache. This nerve sits directly in the back of the nose.
-As far as anything that stimulates nerves, you actually want to desensitize the nerves. Hopefully you can find a Neuro to explain all of this.