You have to tell us what the contest was. Was it a door prize, a puzzle, raffle, work related?
A jeweller or watch shop near you might also appraise it or tell you the market value. The easiest way to sell it is often the least amount you will receive. Without having a history of selling watches, your best opportunity will probably be ebay, as said before. It's not meant to be rude or dismissive. Ebay is actively used by many people here.
You will have better luck with good, clear photos and be as open and candid as you can be. If there is a tiny scratch, mention it. People expect flaws in a used watch. They don't expect surprises. Put a new battery in it and make sure it runs. Tell the story about the watch. Everyone loves a good story. Don't be in a hurry and have fun watching the sale. The last few minutes can be exciting.
Good luck with your sale.
Cheers.