Hi there Paul Anderson,
Thanks for joining us in Apple Support Communities. From what we can tell, you're unable to unlock your Mac with your Apple Watch because an alert appears indicating you must be signed in with the same Apple ID on both devices. We'r glad you're here, and we can provide some guidance.
To start, you'll want to be sure your Mac meets the system requirements for Auto Unlock. Have a look at the 'Auto Unlock' section of this link to check:
System requirements for Continuity on Mac, iPhone, iPad, iPod touch, and Apple Watch
Next, did you sign in with your new Apple ID password on both your Mac, iPhone, and Apple Watch? Have a look at this link to be sure you've taken the proper steps after changing your password:
What to do after you change your Apple ID or password
Next, follow the steps in the 'How to set up Auto Unlock' section of the support article below to attempt to set up Auto Unlock again, and let us know how it goes:
Unlock your Mac with your Apple Watch
If the issue persists, be sure to take a closer look at the suggestions in the 'Learn more' section of the link above.
Take care.