Hi,
The key might be the first line where the iPhone App is turned Off.
This cuts the login to the server much more definitely than just signing out the Apple ID on the iPhone.
Then he (presumed) deselect the iPhone Number on the Mac version before then signing out on the mac version
At the 4th line of re-enabling the app on the iPhone does mean waiting for the Number to Verify.
I would throw in a test at that point
On the iPhone Manually type in the Apple ID used on the Mac.
Sign the Mac version back in and then send from the iPhone.
At this point the iPhone can only use the iPhone Number and the Mac only the Apple ID
If that works add back the Apple ID to the iPhone.
Accept the pop up on the Mac so the iPhone Number is accepted.
Reset Text Forwarding if required.
8:53 pm Tuesday; October 3, 2017
iMac 2.5Ghz i5 2011 (Sierra)
G4/1GhzDual MDD (Leopard 10.5.8)
MacBookPro 2Gb (Snow Leopard 10.6.8)
Mac OS X (10.6.8),
iPhone 6 iOS 11.x and an iPad (2)