What do you do if your nose gets cold on one side when you have something to drink?
This can be caused by nerves or possibly a filling in a tooth. Some fillings can be very sensitive to head/cold and the nerves run from the gums right up to the nasal nerves. If you notice when you have a sinus problem it appears your gums ache, around your nose and under your eyes as well.
Your nose runs in the cold because water is a byproduct of the cold air being warmed, the excess run=s down your throat and out your nose, about the only way to stop it ( as long as it isnt caused by allergies or a cold ) is to warm the air before it reached your nose by wearing a scarf or something to cover it. For water, nose plugs might help.