Solved. Knew it wasn't a 'server setup' error, as we fortunately had a staging version with exact same Acy version (7.9.5), and the signups were perfect.
In my troubleshooting of the live site, i saw an error: "table: acym_custom_zone doesn't exist. "
Checked the mySQL db and sure enough, table was not there. (!!! huh? .. i certainly had not deleted/dropped it... )
Re-installed fresh 7.9.5 version and presto, and it made the acym_custom_zonetable, and the signups are now fine, and no Ajax Request Failure.
If anyone else is experiencing this, check your db and see if the "xxx_acym_custom_zonetable" is missing.
If it is, try a fresh install.
Only cause I can imagine in our case was that doing a previous update, the table was either dropped, or, if it didn't exist in previous version, was not added. ;-)
Joomla 3.10.11
AcyMailing Enterprise 7.9.5
AlmaLinux v8.6.0
MariaDB 5.5.5-10.6.10
PHP Version 7.4.32