You're kind of asking the wrong question. Nothing need be "in" Thunderbird. And "downloading" isn't quite the most useful way to think about it. ;-)

Set up your new account. It will use IMAP unless you deliberately intervene. You can then simply drag-and-drop messages or folders from the old account to the new. (Slightly better, IMHO, to select, right-click and choose "Move to…"; it's a bit cleaner than drag and drop.)

Bear in mind it is being copied down to Thunderbird and then back up to googlemail, so don't try to move too much at one time. So, yes it is "downloading" after a fashion, but I feel that words like "download" and "upload" have a sense of a one-time permanent move from one place to another, and it seems you are wary of this and want the material online and not "in Thunderbird".

When you are happy that everything you want to keep is now in your new account (check it online too, in the gmail website) you delete the old account and close it too, in the gmail website, if you wish.

http://chrisramsden.vfast.co.uk/2_POP_vs_IMAP.html

re :my gmail acct was recently comprised. and i was exposed to a phishing/fraud thing

Did someone actually gain access to your account or did you just get someone abusing your email address without your knowledge?

There are unscrupulous idiots who abuse other peoples email addresses and send emails using it. This is actually more common than you would like and has happened to me as well. Do not respond to these emails. Tell you friends you have had someone abusing your email address and I suggest you create a HTML signature file to use in emails that you send. Tell your friends that genuine emails from you will alwways have the signature, so that they can tell the difference.

After a while the abuser gets bored and moves on to their next victim. Usually, your account has not been hacked; just the email address abused.

To check there is no malware/virus on your computer logging keystrokes or similar, download malwarebytes: https://www.malwarebytes.org/

then restart your computer in Safe Mode and perform a scan using your Anti-Virus software and the malwarebytes program.

Once you have the all clear, then update your passwords.

Do not post personal information & email addresses in public forums as this is typically where abusers find the info. I notice that phillipp had to edit your question to remove personal info.

It is easier to forward emails from one email address to another via the gmail webmail account.

You do not say what type of mail account you are using. If Pop then the emails will be downloaded from the server Inbox to Thunderbird Profile account name Inbox. Ususally, the copy is deleted off the server, but settings can be modified to leave a copy on the server.

If IMAP then you see a remote view of emails on the server. you subscribe to see folders. email headers are downloaded and when you select to see an email, it is retrieved from the server and stored in a temp cache. These emails are not stored in Thunderbird, they are on the server. If you synchronise those subscribed folders, then a copy is downloaded to your Thunderbird Profile, but these folders are constantly updating to match the server. If you delete any email via Thunderbird or webmail, it will be removed from both as these folders are one and the same.

So if you have IMAP then they are stored on the server then you will only have a copy at best.

If you want to get a copy into Thunderbird and delete everything on the server, you need to synchronise your subscribed folders to get full up to date copies in your Profile folders. Then go into 'offline' mode to stop further synchronising, then move all emails into 'Local Folders' mail account. Local Folders is a special mail account - emails in this account are not efected by the server. Remember to create folders to keep your emails organised.

Once you are happy that all emails are in Local Folders account, you can delete the mail in the IMAP mail account and then remove the old gmail account if desired.

You can create a second gmail account in Thunderbird regardless of whether you delete the first one or not:

• File > New > Existing mail account

If you put them in Local Folders then you can move/copy emails back onto server after creating new gmail account OR

simply create the new gmail account and move emails between 

Or Leave the emails in Local Folders to read/store. But these emails will then be only on your computer, so creating a backup would be advisable.