I'm not exactly sure what your question is. In general, make sure to download all your messages from the old IMAP server to your Local Folders account as long as you still do have access to it. Once you lost access, it's too late, unless you do have an offline copy of all your messages in the old account. After creating the new IMAP account you can copy your messages to the new server from Local Folders.

When I restore them, all the emails restore along with the old server settings.

You should restore them to your Local Folders account.

I tried archiving the emails to the local folders, but TB won't let me move them

'TB won't let me move them' could mean anything, you'd need to be more specific about what you're doing, and what actually happens.

In any case, you cannot move messages out of an IMAP account, which doesn't have a server connection anymore. In case you do have an offline copy of your messages, you can put Thunderbird in offline mode, and then move or archive your messages.

Any suggestions on archiving, or better yet keeping my emails before configuring my email client (Mozilla Thunderbird) to connect with the new email account?

See above. Make sure you do have an offline copy of your messages, and create backups of your Thunderbird profile on a regular schedule.

Thank you for attempting to help. I backed up by going to Help>troubleshoot> next to Profile folder I clicked "open folder". Once there I went up one level by clicking "profile" in the address bar, which then gave me the entire folder of settings and emails - I clicked "copy" and then pasted the folder in a safe place. So I have a copy.

I have created a new email account as it needed to be set up to work with an exchange server.

So at this point, the email account I've asked about is no longer connected to any servers. I have archived all the emails from the inbox into the local folders>inbox>archive>2020. However, I haven't been able to move those emails from the archive into the new exchange email account's inbox. They won't copy/paste, drop/drag, nor will they "move".

I also cannot move any of the folders that are in the inbox. I would like to archive them on the local folders. Or, move them into the Inbox of the new exchange server account.

Any direction is much appreciated. Thank you.

I have created a new email account as it needed to be set up to work with an exchange server.

What does 'set up to work with an exchange server' mean exactly? How do you access that server - standard IMAP, EWS, OWA?

I'm using EWS with the ExQuilla for Exchange add-on. I can copy and move messages manually from that account to Local Folders, but I haven't tried uploading anything. The account setup for that Exchange account is somewhat awkward compared to the normal account set-up, and quite a few things don't work at all. Archiving may be one of them.

I also cannot move any of the folders that are in the inbox. I would like to archive them on the local folders.

Try to manually create the destination folder, and then manually copy messages to the desired folder. Finally delete the source folder on the server.

This is very annoying - all the content is locally available - even attachments but still it can be copied to another folder. Is there a way to have it available offline ?