There are many hospitals around Khanom, Nakhon Si Thammarat (863 beds), Surat Thani (660 beds), Tha Sala (60 beds), Sichon (120 beds) Koh Samui (60 beds) with a helicopter service and a boat ambulance but also "Khanom" has a small hospital (30 beds) in town next to the police station.

http://en.wikipedia.org/wiki/List_of_hospitals_in_Thailand

Recently (2014) Dr. Ning, a English speaking dentist opened her new practice in Khanom, opposite the Wednesday market. She is open Saturday and Sunday 8:30 - 16:30. Call her office @ +6686 517 9223.

If you need urgent dental treatment during other days you can go to Surat Thani hospital. 

Also there is a New Doctors Surgery. At the Cross roads, opposite school. Senior doctor from Khanom Hospital.