Thyroid cancer can potentially cause hoarseness or difficulty in speaking in a normal voice, trouble swallowing or breathing, pain in neck/throat and swollen glands. On the website I looked at, they didn't list coughing. However, coughing can be a sign of acid reflux disease- so you may want to consult a gastrointestinal specialist if you suspect that might be a problem for you.
Yes, acid reflux seems more appropriate...you can test this yourself before seeing a doctor by going to the drug store and getting Prilosec OTC, take one pill 1/2 hour before breakfast and dinner for two weeks, and see if it helps. If the coughing gets better, you probably have gerd.