Yes, it isn't about signing into and out of icloud. It's about using the same APPLE ID (nothing to do with icloud) entered into FACETIME as the APPLE ID for FACETIME...and for iMESSAGES...The Apple ID must match in 2 places on each device: Facetime...and Messages...The system settings/preferences iCloud panel it not connected to this. Just like neither of these are connected to the APP STORE. That could be ANOTHER Apple ID if you like, for purchases...
Facetime and Messages are not part of iCloud. :-) Also, I would sign out and put the APPLE IDS into FACETIME, THEN if you can (you may not need to), sign out of Messages and sign back in with the same APPLE ID.
Also, check that handoff is on...(This IS the same iCloud APPLE ID required, I think)...and that WIFI CALLING IS ENABLED ON ALL DEVICES...AND THAT WIFI and CELLULAR are turned on for data on all..and WIFI assist is on...Careful if data use is an issue.
Frankly, you may find this is all more trouble than it's worth. I have it on, but it's basically annoying as heck. Cumbersome, loud, kinda confusing...But, let us know if you enjoy it