I am one of the unlucky ones as well. My main concern with how long it will take to fix it is screen burn-in from the Apple logo. After trying multiple button combinations, I found one that works to power the watch off:
1. Hold the Digital Crown and the side button together until it powers off.
2. As soon as it goes to black, and before the Apple logo comes back, continue holding the side button, but let go of the Digital Crown.
3. When the Apple logo does appear, hold the crown down once again.
4. As soon as it’s back to black, let go of both buttons.
When I spoke to support, they made it seem that this would be a software only fix and not a swap. So it’s best to follow these steps, preserve the display and not risk a burned in logo.