Hello stephenfromcenterville,
If you signed out of iCloud on your iPhone, your Apple Watch should have also been signed out: Sign out of iCloud on your iPhone, iPad, iPod touch, Apple TV, or Mac - Apple Support
When you sign out of iCloud on the iPhone that's paired with your Apple Watch, you're automatically signed out of iCloud on your Apple Watch.
If you haven't already done so, let's use the steps below to make sure you are still signed into iCloud on your iPhone:
1. On your iPhone, iPad, or iPod touch, go to Settings > [your name].
If you don’t see [your name], tap “Sign in to your [device],” then enter your Apple ID and password.
2. Do one of the following:
• iOS 16, iPadOS 16, or later: Tap iCloud, then turn on each app or feature you want to use. Tap Show All to see more apps.
If you are already signed into iCloud using your new Apple ID on your iPhone, does that still persist after restarting your Apple Watch? Restart your Apple Watch - Apple Support
Let us know, and we'll keep moving forward from there.