This is most odd! Are the fields registered via PHP code, or are they created via the standard database UI?
Have you tried setting up the website on a local MAMP server? Perhaps there is an issue with POSTING data
My first through is that there is some conflict going on either from another plugin or from the theme.
Did you deactivate 100% of all plugins AND change theme to default? Can you make sure the default theme did not contain any custom functionality?
I would setup on a local install and start fresh.
1. Re create the field group (perhaps even import / export)
2. Create some new posts and see if it works
3. Install your plugins 1 by one until the issue starts again
4. Install your theme.
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