Is money earned from selling wristwatches in this manner, where the customer is informed of winning a gift but is required to pay a delivery fee, permissible (halal) or forbidden (haram)? And is this method legitimate?

The trick you are employing to unlawfully acquire people's money by deceiving them into believing you are giving them a gift, then demanding a delivery fee for it, is prohibited fraud and deception. The money earned from it is illicit. "Whoever cheats us is not of us." Deceit and trickery lead to Hellfire. Scholars have explicitly stated the prohibition of all forms of deception (khalabah) in sales and other transactions, and the prohibition of lying and deception that put the buyer in an awkward position. It is incumbent upon you to repent to Allah, cease deception and lying, and pursue legitimate means in your transactions.