Hi there! So first I would highly recommend you guys separate your iCloud accounts. Each should have their own iCloud account to avoid issues such as these. Now if the reason you would like to share accounts is to share purchases then you can set up Family Sharing and still have separate accounts.
My suggestion would be to first decide who will be keeping the current iCloud account.
Then, log off the person who is getting a new one and when it asks you "do you want to delete content on this iPhone?" or something along those lines choose "no, i would like to KEEP"
After that you can create a new one through that same screen and it will ask you if you want to merge your existing information and you will click "merge".
Finally, after all the above is done make sure to check in your settings in the sections iMessage and FaceTime to ensure that the NEW Apple ID is logged and not the old one.
See if that helps with the messaging issues.
Let me know,
Jonathan A