Hello Lonelymelody,
You have a few choices. If you write "Dr. Mary Smith" you could be a medical doctor, or a dentist, or even a Ph.D. If you write Mary Smith, M.D., you have left no doubt that you are a medical doctor.
However, if you were writing about something that has NOTHING to do with medicine - making a hotel reservation or writing to follow up on a missing order for boots - there isn't a very good reason to include it. Nonetheless, you worked hard for your degree and you can show that you've earned it by including it.
Just do NOT include both Dr. Mary Smith, M.D.