I remember a while ago I came across the same problem, I found out that PHP never cleaned the PHP session folder of old files. At some point it contained over 2.5 MILLION session files which made the site extremely slow.
You should take a look at the folder where PHP is saving the session files and see if PHP is correctly garbage cleaning the folder.
If that's also not the case, you may want to consider paying someone to look into it. It's really hard for us to figure out what's going on without having access to the server.
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