I wouldn't change the old account settings, just add it as a new account for the IMAP details.
Your old POP account should be left as is, perhaps just turning off the 'Include this account when checking for new mail' in the account prefs. You could move it to a 'local mailbox' if you really want to archive it. Use the + icon at the bottom of the mailboxes & add a new mailbox 'on my mac'. Take care if you do move (instead of copy) it, you don't want the other mac repeating the steps or just removing the messages from the inbox. POP shouldn't sync changes but it might try. Make sure both are inactive, remove the passwords (in /Apps/Utilities/Keychain Access) to be 100% sure it won't connect to the server.
Turn on IMAP on the gmail settings page.
Create a new account to handle the IMAP settings in mail, add the settings that Google provides.
Add another new local mailbox to store the offline archive.
You can add a rule to Copy to 'local mailbox name here' to duplicate messages into the local mailbox. That will give you an offline copy. Chances are that you may not need it because IMAP caches locally anyway. See how it works. You may want to create some folders on the server to better organise the mail. They will then appear in each client using the IMAP settings.
Some Gmail tips.
'Mail account prefs > Advanced' set the IMAP path prefix to '[Google Mail]' (no quotes but keep brackets) to remove the 'Google mail' container in the sidebar.
Select the 'sent mail' for the new Google account in the sidebar and do 'Mailbox > Use this mailbox for > Sent'
It will insert the mailbox into the main sent items container. Repeat for Drafts & Trash. I can't really remember if Spam should be left to Google or Apple mail, I think it should be left as is to let Googles filters dump it into the spam folder.
Hope this helps
Re:co