The dry cleaners you mean is called My Dry and is located on the left as you head towards Migros from Calis, quite expensive.
There is another one considerably cheaper in Fethiye, with the post office behind you take the road opposite, take the first turning right by the watch mender, you will come into a small square, the cleaners is on the right side you can't miss it as he usually has wedding dresses hanging outside.
For a good and reliable tailor go to the lights from the mosque, turn left onto the main road them immediately right by the cash machines, take your first left, on the right side you will see a passage way, he is on the left at the bottom, say Karen from the card shop sent you