We always want all our features and options to work for you, cratervoid. We are here to help.
Since this issue started have you unsubscribed from Integrated Messaging and then re-subscribed? We ask because this refresh process typically fixes this syncing issue with computers.
If you have not done this already, you can do this from the Message+ app on your phone. Tap the Menu icon on the upper left of the screen in the Verizon Messages (Messages+) app, tap Settings, Scroll to and tap Unsubscribe.