Turning off Find My on Apple Watch 6 but keeping it on for iPhone?

Turning off My Find would be a bad idea. If you ever loose your watch you will not be able to track it or even delete it.

Why not just turn off Location sharing?

On your watch>Setting>Privacy>Location Services>Share My Location>Toggle this off

Greetings, Littlegreendog.

Welcome to the Apple Support Communities. It sounds like you would like to know how to remove your Apple Watch from iCloud. We can certainly understand wanting to use your new Apple Watch instead.

Do you still have access to your Apple Watch? If not check out:

“If you can't access your iPhone or Apple Watch, or if your Apple Watch isn't paired with your iPhone and you need to turn off Activation Lock:

1. On a computer, go to iCloud.com and sign in with your Apple ID. 
2. Go to Find My iPhone.
3. Select All Devices, then click your Apple Watch.
4. Click Erase Apple Watch. Select Next until the device is erased.
5. Click the remove button  next to your Apple Watch.” About Activation Lock on your Apple Watch

If so, then take a look at: Unpair and erase your Apple Watch

We hope this helps! Don’t hesitate to respond if you have questions.

Warmly.