New Host is using the newer Exim format. You can tell which format is used by looking under Server Status, in Cpanel or WHM.
old eMail method uses mbox (exim-4.52-7)
new eMail method uses maildir (exim-4.52-15_cpanel_maildir)
"After "MailDir" conversion, individuals logging into Horde WebMail from
main user of control panel will no longer have opportunity to access all
other mailboxes under same domain name. This resolves privacy concerns
but for some, the previous ability to access other mailboxes was very convenient."
In other words, whether using IMAP or not and with the 'maildir' format, you can only see the eMails for the Mailbox account you have logged in as.
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