I don't have any particular shop to point out, but generally speaking
Cesme town itself is a "compact" and "bustling" collection of Souvenir, Leather, Carpet, Clothing, Footwear and Jewelry stores.
What to buy: You can buy anything according to your needs and taste.
What to pay: Expect to pay much less than in a European country, and be prepared to make a bargain.
Cesme's prices are much cheaper than Istanbul. There are a whole assortment of Turkish products to buy. When buying carpets offer them half of what they ask and settle for 35 to 40 % of the retail price.