I thought about trying Thunderbird for a while, but we’ll give the new Mail.app a chance first. To transfer my email, I first created folders (on my eMac) called TempInbox, TempSent, TempDrafts, and TempTrash. I then took all the mail from the Inbox, Sent, etc and transferred it into those folders. This way all the emails are in one folder that I can transfer and import. I took the “Mailboxes” folder, ~/Library/Mail/Mailboxes, and transfered it to the iMac.
On the iMac, I sent up my account as it wanted, and then went to File>Import Mailboxes. Then I choose the Mailboxes folder that I transfered. Dragged the emails from the Temp folder’s and now my email is back in place.
I initially had a message saying “Mail can’t verify the identity of mail.xyz.net”. This would show up each time I opened Mail, but by clicking Show Certificate and checking the box Always trust, I don’t see this anymore.