If you guys are coughing, you really need an air purifier. You have no idea the difference it will make for your breathing.
There are a few threads about these machines and you can get some ideas.
I don't want to be an alarmist but if you get a problem related to breathing with birds around, you will have to give your birds away

I live in dread of this because giving away my birds would be like dying

Please look into air purifiers!!!