What I have may be similar or the same as what you have, so let me describe my situation and then you can match or modify it.

I get text messages on both my iPhone and Mac. When I delete a text message on my iPhone, it is gone, but it still remains on the Mac forever. The only difference in settings that I can see is that on the iPhone under iCloud settings in Settings, "Messages" is set to "off". This seems to allow my Mac to get messages and to keep them when they are deleted on the iPhone. There do not seem to be any other settings on the Mac that relate to Messages and iCloud (not even in Settings>Internet Accounts>iCloud).

One downside to your free iCloud account being full is that your iPhone will not have a current back up.

On a Mac since 1984.
Currently: 24" M1 iMac, M2 Pro Mac mini with 27" BenQ monitor, M2 Macbook Air, MacOS 14.x; iPhones, iPods (yes, still) and iPads.