Possible fix?
Apologies if someone's already pointed this out, but I wanted to pop back into the thread to update on a fix that, for now, seems to be working.
Recap:
1. I have six family phones on my account, each logged in under its own unique Apple ID for quite some time before IOS 12 and the cross-texting issue. The "get your own Apple ID" suggestion was not a solution for me.
2. Only three of the six lines in the family were having an issue.
3. We discovered that, despite the affected three phones displaying a unique Apple ID (Settings>>select Apple ID under the picture on the first screen), when looking at the "Send & Receive" screen in the iMessages app (the screen that allows you to select the phone numbers used by the phone), the three impacted phones were all still displaying the same primary Apple ID (the one on my phone from back in the pre-Family Sharing days when we all shared one Apple ID).
4. The other three non-affected family phones correctly displayed the user's unique Apple ID inside the iMessage app.
5. From the iMessages setting, we logged out the two affected phone and re-logged them in under their correct Apple ID.
6. The problem seems to have resolved itself for now.
So, even if you switch your phones to unique Apple IDs (or already had them set that way before IOS 12), try double-checking:
Settings>>iMessage>>Send & Receive. Make each phone is in fact on its individual Apple ID inside the Messaging app.
Hope this helps.
p.s. When we switched to unique Apple IDs some time ago, none of us individually set the iMessage "Send & Receive" screen. For some reason, two of the six phones didn't switch over (mine never changed as it was the original primary/shared Apple ID).