I stopped receiving new texts in the app on the watch. I also could not send new messages from the app on the watch.
That said, however, I was still receiving new notifications, and could reply via the notifications.
I first tried clearing the cache, then the data, however it didn't change the behavior. This is when I uninstalled the app from the watch. When I tried to reinstall the app on the watch, it is no longer listed in the Play Store on the watch.
To make matters worse, now that the app is no longer on the watch, I no longer receive notifications on the watch for new messages. I have verified that Wear OS -> Notifications -> Message+ , is enabled to send notifications.