Hello there, dbratnyc!
Thanks for reaching out to the Apple Support Communities! It sounds like you have an issue with your Messages not syncing to your computer properly. To start, please make sure you have Messages in iCloud enabled on all devices: Access your messages on all your Apple devices
How to forward SMS/MMS text messages from your iPhone to your iPad, iPod touch or Mac -- On your iPhone, go to Settings > Messages > Send & Receive. On your Mac, open Messages and choose Messages > Preferences > iMessages. Make sure that both devices are signed in with the same Apple ID. Then, on your iPhone, go back one screen and check Text Message Forwarding to make sure your Mac has been selected. If it is, deselect your Mac and restart your iPhone. Once it's back to the Home screen, turn Text Message Forwarding back on for your Mac and test the issue.
If Messages continue to not update across your devices, we would recommend reaching out to Apple directly by checking out this link to Get Support and they will be more than happy to proceed with additional steps.
Best wishes.