Actually, my ds did cough while teething, and ran fevers (yes, we did confirm with dr. visits that was the cause of both). Spit/drool would constantly run down his throat, and he didn't know how to clear it out properly at the time.
Fluids are a must. Also, if he is teething, have you tried the baby orajel? When ds wouldn't eat because of teething pain, we used the baby orajel (in addition to the tubes, they also have some that come in q-tip type things. Shap the end on one side, the orajel flows to the other tip. Easy to apply.). We would also let him chew on a frozen washcloth. That would help numb the sore spot, and get him a little dose of water at the same time as the washcloth thawed out.
Good luck!