Paired? That's a Bluetooth thing. Bluetooth has nothing to do with whether you get notifications or not. On the old phone, go to Settings and turn off the notifications you don't want.
Is your Watch paired with a new iPhone? If so, it's signed into your AppleID. It gets notifications from Apple over wifi, not from an iPhone over Bluetooth.
Is your Watch paired with a new iPhone? If so, it's signed into your AppleID. It gets notifications from Apple over wifi, not from an iPhone over Bluetooth.