You can use Thunderbird to move emails from the old to the new folders. If you are having trouble, it could be one corrupt email. Just do small batches until you find the culprit.
Gmail uses tags. But traditional IMAP is folders
-> Sent Mail to Sent Items
-> Inbox to Inbox
Then, what is left will be in "All Mails"
-> Remaining mail in "All Mails" -> Archive per year
Q: what about mails that have multiple tags?
If you are running old & new system in parallel for a while, you can invite the other account as an attendee