How to switch to another mail service provider on Mac Mail app?

Just create/add a new account to Apple Mail:

Add email accounts in Mail on Mac - Apple Support

Regarding your old account you will likely have to export them or something to save them since I would guess that after some time after not using the old account, your messages will be removed from the servers and it IMAP they will likely disappear from the Mac. You have to check with your email provider of what to do with old emails messages.

Ok, in that case I would try the following:

Create a new mailbox (Mailbox > New Mailbox). In the resulting dialog box, select "On My Mac" as the Location.

This will create a local mailbox whose contents will be stored on your Mac.

You can add (create) mailboxes within this mailbox to match each folder you may have in your Exchange account/mailbox.

Then copy emails from your existing Exchange mailbox folders to the new mailbox folders that are "On My Mac."

Try this on a single folder and just a few emails to verify that it works properly. If you are satisfied with the results, then copy everything you want to keep.

Since your organization is retaining the same domain & email addresses (just changing hosts), it would be best if your organization could arrange with the new provider to migrate all mail account(s) from your current service provider to the new platform. Server- based migration would preserve everyone's folders & emails. Users would only have to modify the account settings in their email client to point to the new server once migration is complete.

Exchange is an IMAP-based mail server, so the mails reside on the server.

You could try exporting your Exchange folders to .mbox files. Do them one at a time, as .mbox does not support subfolders. In the Mail app, select a folder, then do Mailbox > Export Mailbox. Select a location and then click Choose. There is no guarantee that it will be any faster than the copying I suggested earlier.