Session mismatch, no connection

Hello,
We need to backup tables to transfer the website (about 45000 posts, 3 millions comments etc) to another server.
We use an OLD php 5.6 release, and the point is that current hoster does not want to do nothing! We have NO Server access, just WP admin rights.
That is why we are looking for a WP plugin.

First we were told to install the “old” phpmyadmin version (because of old php version). Got it, and when we click to “Enter hpmyadmin” we get this:

“Session mismatch
Now, go back to Dashboard-PhpMyAdmin page, and then click Login phpMyAdmin button again. If you still have problems, open ticket at Support pages, probably something breaks a normal page-load in dashboard.”

Even after page reload, same problem. I add that we could enter ONE time, only. After, always this problem.

Any idea? The point is we couldn’t find any other plugin that allows a BY TABLE download… Full DB is too big to download it in one time, so we need table by table. So we do need your plugin ??

Thank ypu!