IMAP or POP?

If POP then your messages are already stored on your computer and will remain after closure of the account. However you will get irritating messages when Thunderbird tries and falls to connect to the old server. In your position I'd move the messages into a working account, or Local Folders.

If IMAP then you need to act. The messages are stored on the server and will vanish when the server becomes unavailable. You need to move all the messages you want to keep to an account you'll still have after the changeover. So you could move these messages to a working account, such as one of your Gmail accounts. Alternatively you can move them to the Local Folders account in Thunderbird.

In both cases, moving precious messages to Local Folders will do the job. It's a bit quicker since you're copying via download to your own machine. Copying to an IMAP-connected account also involves an upload to the server, so might be a reason to avoid using the Gmail accounts, given the number of messages you have to rescue.

Thank you Zenos. Both are IMAP accounts. If I want to transfer to 'local folders' can I just move the folder there or do I have to move each email individually? Is there a limit of the number you can transfer at a time? Thank you.

Because of the way IMAP works and the historically tardy nature of anything BT. I would actually recommend exporting the mail using the import export tool and importing it again. While what you want to do can be achieved through drag and drop of mails such actions are prone to dropout and freezes as well as plain failure where some mails just disappear.

Use the tool to export everything in the BT mailbox as it will be the only one affected by your move. Right click the account name in the folder pane and select importexporttools then export all the folders with structure. Take a special not of the location you are saving to as you will been to find it again. You could create a folder on your desktop beforehand to make it simple.

Then right click the "Local Folders" heading in the folder pane and select importexporttools > import an MBOX file in the next dialog select the last option that ends with (also in subdirectories) this will bring the exported mail back into Thunderbird. This process breaks the link between the mail and the mail server that is inherent in IMAP without the limitations on volume imposed by the IMAP protocol and your anti virus which will interfere.

Get the import export tool here. https://addons.mozilla.org/en-US/thunderbird/addon/importexporttools/

Thanks so much for that Matt, quite agree with you about BT drop out and freezes. Trying out your tool now. Cheers!