You should be able to transfer your e-mail hosting to another provider and keep your ISP and their web site hosting.

Are you currently using a Microsoft Exchange Server setup, or just POP3 e-mail?

Possibly useful link:

https://www.techradar.com/news/best-...ting-providers

https://www.rackspace.com/en-us/email-hosting/webmail

https://products.office.com/en-us/ex...e-online-plans