Same issue for me
Not badges.
Any notification on my iphone 13 appears on Garmin watches - all three watches. Mail, Ring alerts, text, etc. When cleared on iphone, it no longer clears on Garmin even though sync'd. It requires a lengthy manual clearing on watch. Turning watch and/or phone off/on does not clear. Sync through Connect app does not help.
Next?
Probably a problem with memory full in the watch?. When you connect your watch to your PC you can see all folders like an external drive. You should delete those old notifications from there but I have no idea what folders to look in :(