IMAP is definitely the way to go for this type of thing. Sync'ing your email or "leave copy on server" can get messed up very easily.
Doesn't matter what email you have today, you can forward it somewhere else. The "somewhere else" should be a provider that supports IMAP along with POP.
This will allow you to connect whatever email client you have to your email repository which will stay on the provider's server. The key is to have enough space to keep all your email on the provider's server. I use Gmail because of the ever growing space allocation they have. So for the past 7 years they have allowed my disk quota to grow with my usage.