Hey again!
I know you’ve signed out/in, etc. already, which are good steps.
Do you have iMessage enabled on other devices? And if so, how are those responding?
First, I’d check my phone settings:
Settings > Messages > Send and Recieve.
Settings > Messages > Text Message Forwarding.
I’d just make sure both your Apple ID and phone number are showing there.
Likewise that in TMF, that it’s turned on for your Mac.
It should show the name of your Mac. On your Mac, you can see the name of it, as your other devices see it in:
System Preferences > Sharing > (My Mac Name). (I’ve even found that disabling TMF from iPhone, changing the Mac name, re enabling TMF is a good step.
Likewise, sometimes changing the name of your phone, and re connecting your Mac:
Settings > General > About > (Phone Name).
Then restart both, (And/or other devices), and test.
I just said those, because they are simpler tests to try.
However, if you really want a good test, which may tell you the cause, leading to a solution:
Test while your phone is only connected to WiFi, and Cellular Data.
Test in a new admin account.
Test in Safe Mode.
Test while both are connected to another network.
Another thing to think about is:
Do you use iCloud to store/sync messages? And if so, how is your iCloud Storage?
I’ve seen a lot, there could even be one recipient in your group that’s rejecting the error and jamming up the system. Although unlikely.
If you can answer those questions, hopefully will find a solution, let me know how it goes!