You probably wish you had grabbed everything before converting.
If you do as they suggest, mail may start going to register.com again. I don’t think you want this.
You may want to try the following instead:
Do an nslookup to find the IP address for webmail.register.com.
Add a host entry on your local system for mail.yourdomain.tld using the IP address found above (this will provide you the function of the CNAME DNS entry they are suggesting).
Reconfigure your mail client back to the way register.com wants it.
Do whatever you need to do to get your address book (I have no clue).
Remove the hosts entry and convert your mail client back to DH.
This should work as long as register.com will not be picky about not having an MX entry for your domain pointing to them.
If you do have to create the DNS MX entry, make sure the priority is lower than the priority for DH. If, for some reason, DH mail goes out, mail will go to register.com. MX entries tell other systems where to send mail.