Solution No 1:
Log in to your first cPanel of example.com (for website)
Upload all the files here. Then, click "Simple DNS Editor" in the same cPanel. Under "Add an A Record" Name:mail.example.com Address:your_mail_server_ip address (ask your mail service provider for this)
And click "Add a Record".
Once this record added.
Goto "MX Entry" Choose "Remote Mail Exchanger" Click Change.
Then delete the previous MX record. Now add mail.example.com in Destination, Priority is 0 (zero). Click "Add New Record"
That's it.
Solution No 2:
Upload your contents in Cpanel used for the website.
Now, log in to your other cPanel used for mail service.
Go to "Simple DNS Editor" under Add an A Record
Name example.com address: A record IP address of your website server.
Create an email id in the email server (cPanel) and start using the mails and website in different servers.
Hope this helps you.
Solution No 3:
If this confusing you. contact your hosting provider with the MX record and its IP address.
Ask them to update that MX record from their end.