Thanks for the update, thutty! We appreciate you providing that information.
Since the previous steps have not solved the issue, I would next recommend turning Text Message Forwarding, and iMessage, off on both your iPhone and your iMac, restarting both, then setting everything back up first on the iPhone, then on your iMac.
Sign in with your Apple ID - Apple Support outlines how to sign into your Apple ID for various services, including iMessage, on both the iPhone and the Mac. You can use those same steps to sign out, as well. Likewise, the following steps from Use Continuity to connect your Mac, iPhone, iPad, iPod touch, and Apple Watch - Apple Support will show you how to set up Text Forwarding on your devices, and you can use those same steps to turn it off, as well:
Have a lovely day!
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.