DNS change email error

[Patato]

Hello, I used to work with a hosting company, and my domain was connected there, and I was using the email service. I switched to a new hosting company, and when I changed the DNS settings, I lost the emails in my mailbox. I set up the same email address with the new hosting company, but the old emails did not arrive. Can you help me with a solution? The old hosting company didn't have cPanel, so I can't access the files by logging into the panel. They are suggesting that if the old hosting company has not been deleted, we can retrieve the history and transfer it. If I can get these files, how should I proceed? Thank you in advance.