Yes. Especially because nasal congestion can cause post nasal drip (where it drips down the back of the throat) That post nasal drip can cause the tonsils to enlarge because of irritation.
Is your house dusty, musty or moldy?
I would suggest looking around for an air purifier that ozonates the air. There are a lot of manufactures, but I prefer *Living Air. I believe their machines are far superior.
Of course make sure both your children have a clean diet of lots of raw fruits and veggies, whole grains and absolutely no artificial ingredients like dye and flavors. I would also cut out all dairy products.
Is the air conditioner brand new or used? They are famous for a build up molds and dust in their filter. They need to be taken apart regularly and clean well.