Hi Amber,
I could be completely wrong, but I am inclined to agree with your suspicion of a nasal infection. Sounds like it to me. I would drink plenty of water (an nothing else, no milk, coffee, or even juices, unless you make them fresh, and drink them fresh), and use a drug free nasal wash. Xlear is a good one, and sold many places these days. It is all natural and helps make your nasal passages slippery so your sinuses can drain.
Another really helpful way to aid sinus drainage is to practice relaxing daily. The tension you store in your muscles, especially your neck, tend to bind things up. If you spend time releasing tension, you may notice a drastic improvement.
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You might be exposed to smoke without being aware of it. Smoke can enter a room from under a door or a crack in the window frame or through a central ventilation heating and cooling system. This could be a cause of your cough. On the other hand, given your previous experience with odd smells, this could be another of those.
The cough with "bad chest pain" could be causing one or more of the nerves coming off your spine, being compressed by the high pressures generated with a cough. Such compression of nerves can occur with collapsed or arthritic vertebrae. You may want to see your doctor to have x-rays of the bony spine to see if this could be the problem. If you do have such x-rays and are still coughing, you should also have a chest x-ray at the same time.