If you want to keep the emails and use just one account you'll have to pay for more storage.
On a Mac, you could just move the messages to a folder that is located "On My Mac"...but other devices would not be able to access them.
My "free" solution would be to set up an free account somewhere (like Gmail, which has 15GB of storage), create folders at that account for sorting (if you wish). Then, add the email account to your iPhone and "Move" the iCloud messages to a folder or folders (the "Inbox" if you wish) on the Gmail server...you can do this on the iPhone. Be sure to check that your iCloud account remains the default one after you've added the new account.