@turquoise-- Your doctor is aware that you're pregnant and I'm assuming that he or she prescribed a medication that's safe to take during pregnancy. If you have doubts about that, get a second opinion with another doctor.
I had bronchitis during my first pregnancy. I resisted medications and did my best to recover on my own, with the help of remedies like a humidifier and herbal tea with honey. The remedies helped a lot but my bronchitis got worse and I actually ended up having to take antibiotics.
My doctor told me that when it comes to pregnancy and medications, they won't prescribe them unless the benefits outweigh the risks and in my case, they did. An untreated infection is much more dangerous for a baby than medication, especially at this stage in pregnancy.
If I were you, I would use a humidifier, sip on hot liquids and if I still need something for the cough, I would go ahead and take a low dose of the medication my doctor gave me.