Hello,
Thank you for your query and we are so sorry about the trouble this must have caused.
We have added your account access in the sensitive data section.
Assuming that the Free and PRO versions are already installed on your website, please reconnect the website with the correct account to fully utilize all the features.
Please check if you are running v3.0.84
of Rank Math PRO and v1.0.241
of the free version.
The PRO version is built on top of the free version to keep the plugins robust and lightweight, so you need to keep both installed and activated to use the PRO.
Here’s our guide about this topic: https://rankmath.com/kb/should-i-delete-free-plugin-after-upgrading/
Yes, you can overwrite the previous plugin, and all settings will be ported to the updated plugin.
Please reconnect the website to your Rank Math account and then clear the Rank Math Transients by going to Rank Math > Status & Tools > Database Tools > Remove Transients:

If the issue persists, we might need to take a closer look at the settings. Please edit the first post on this ticket and include your WordPress & FTP logins in the designated Sensitive Data section.
Please do take a complete backup of your website before sharing the information with us.

It is completely secure and only our support staff has access to that section. If you want, you can use the below plugin to generate a temporary login URL to your website and share that with us instead:
https://wordpress.org/plugins/temporary-login-without-password/
You can use the above plugin in conjunction with the WP Security Audit Log to monitor what changes our staff might make on your website (if any):
https://wordpress.org/plugins/wp-security-audit-log/
We really look forward to helping you.