This is what I'd do.

1. If some sort of folder hierarchy exists for the old account, manually re-create that folder hierarchy at the new account.
2. Then copy over messages from the old to the new account folder by folder.
3. Select a batch of messages in a folder on the old account - right click a selected message - Copy To - select the destination account and folder.

Start with a small no. of messages per batch, and see how it goes. Do not attempt to transfer a large no. of messages at once. Mail servers are not made for mass transactions, and email providers may limit the amount of data allowed to be transferred within a certain time. So take your time. Make sure to use 'Copy', and not 'Move', so you won't lose any messages in case something goes wrong. Once everything has been transferred, you can delete the messages in the old account.

Thanks...

Unfortunately, I already started doing what you suggested (relatively small batches) but I have been using the "Move" command. I'm trying to move messages from a Yahoo mail account which I have configured in Thunderbird, to an Outlook account, which is also configures in Thunderbird.

To test, I selected a few messages, then right-clicked to bring up the menu. I see an option "Move to <my folder name> again..." I'm confused, as I've tried this for messages that I'm 99% sure I had not yet moved.

I could just go delete the destination folder (the new one I created in the destination mail account to match the folder structure of the source account). But now I'm afraid I might lose some messages if I have really moved them and they no longer exist in the source mail folder.

Another confounding observation. I tried moving a few hundred messages (not all at the same time) in a single session where I did not close and reopen Thunderbird. The message count appears to decrease as expected: after moving messages, the message count of the source folder decreases.

However, after closing Thunderbird and then reopening it later, I see that the message count is not the decreased value but the original message count (before I started moving mail messages).

Does any of this sound familiar?