iMessage on Computer/Phone Glitch?

Hi there, davidjosephl!
Thanks for reaching out to the Apple Support Communities! It sounds like you are having issues with a Messages on your Mac and notifications on your iPhone and I am happy to help out! First, I recommend you sign out of Messages on the Mac if the issue is only with iMessages and then sign back in. If the issue is also with SMS messages, you’ll need to check out this article to Use Continuity to connect your Mac, iPhone, iPad, iPod touch, and Apple Watch. Specifically, use this section to be sure Continuity is set up properly for the devices to communicate, which will help with the sending issue and potentially with the notifications:

Set up SMS and MMS messaging

Use this feature with any Mac, iPhone, iPad, or iPod touch that meets the Continuity system requirements. Make sure that your devices are set up as follows:

• Each device is signed in to iCloud with the same Apple ID.
• On iPhone, go to Settings > Messages > Send & Receive. Make sure that the Apple ID at the top of the screen is the same Apple ID that you're using for iMessage on your other devices. Add a check to your phone number and email address, so that you can be reached by iMessage at both. Do the same on your iPad or iPod touch.
• On iPhone, go to Settings > Messages > Text Message Forwarding, then choose which devices to allow to send and receive text messages from this iPhone. If you're not using two-factor authentication for your Apple ID, a verification code appears on each of your other devices: enter that code on your iPhone.
• On Mac, open Messages, then choose Messages > Preferences. Click Accounts, then select your iMessage account. Make sure that the Apple ID shown here is the same Apple ID that you're using on your other devices. Add a check to your phone number and email address.

Let me know the results and have a great day!