Hi, I can certainly share my experience. When we traveled to Fethiye, we also used the airport shuttle. Generally, it's convenient, no hassles with cabs or public transportation. Prices depend on the company and the type of car, but on average for economy class one way you can count on 30-40 euros. Personally, we used the services of KiwiTaxi. They met us at the airport at the agreed time with a name badge, helped us with our luggage and took us directly to the hotel in a comfortable car. Everything was quick and no problems. Here is a link to their website, where you can immediately arrange a transfer:
https://kiwitaxi.com/en/turkey/fethiye-dalaman-airport. Hope this helps you with your transportation planning, have a great vacation!