Click on the “profile” icon button on top right of nav bar and navigate to “Issue Navigator”:
Type the following into the search bar in Issue Navigator: project = "???????" AND watcher = currentUser() and hit enter. Replace ?????? with your project name
Then choose the “Tools” button on the top right and select “All 124 issues”
Then click the check box next to “Summary” to select all issues and then press “Next”
Then select “Stop watching issues” and hit next.
For anyone reading this issue now. It looks like they have slightly changed how you can achieve this, from the search box in top right of the navbar, you want to click on advanced issue search then click the text next to the search button that says "Switch to JQL". You are then able to run the search as described above.
You can achieve this using bulk transition.
If you have "bulk edit" permission (it's a global permission), then you can run a filter for "project = X" (or even "I am a watcher" or whatever else), bulk edit the results and remove yourself as a watcher from everything.