It wouldn’t be a dreamhost glitch if you lost your old mail.
With POP the messages are pulled from the server and reside locally on your computer / phone. With IMAP the main folder resides server side with local copies. Thus as you say, when you delete from one device, the folder on the server is updated which updates other devices.
That said it shouldn’t be a problem to switch, just make sure you have backed up the local messages on your comptuer before you start. They may disappear from your phone. Once IMAP is up and running you can open your local folder and copy to an IMAP folder and then you will have access to those on your phone again.
IMAP is a better answer these days, with everyone using a computer and smartphone, but they key here is… as always… make sure you have a backup before you make changes.