Back up and recover Outlook / Express
Search your old computer for files ending .dbx. These are your emails.
There should be Inbox.dbx, Outbox.dbx, Drafts.dbx etc.
There must also be Folders.dbx, this is the data telling OE how the emails are compressed.
Save all these dbx files, then you can either import them into OE on your new computer, or overwrite the files of the same name in the new computer.
Look for a file called (name).wab, this is your address book, and a folder called Favorites. Save these and either import or overwrite to transfer them.