I've had a pretty frequent dry persistent cough. Should I be on antibiotics?

No... A non-productive persistent cough is likely due to bronchospasm from any cause, gerd, post-nasal drip or a combination of these factors. It is not due to infection so antibiotics would be of no benefit. If you smoke, you need to quit to decrease the airway inflammation and decrease your risk of copd, malignancy, etc.