So! As it turns out, codeine packs quite a punch. I've spent the past couple evenings essentially passed out due to cough syrup. Surprising, because I'm not usually that sensitive to drowsy-making things. I guess my decision not to take any cough syrup during the day was a good one, both for my productivity at work and being able to stay awake to prevent the children from burning the house down.
On the plus side, the theory that stopping the cough will help the irritation calm down seems to be panning out. I've been regular with the inhaler as well, and while it's not gone completely, I'm coughing a lot less, and it no longer feels like an elephant is standing on my chest - more like a small pony. Progress!