Are you using IMAP or POP email?

I have no idea. I'm not very tech savy. Where would I find that information?

Copy all the messages you want to keep to the Local Folders account in Thunderbird. Then the type of account is immaterial.

However…

The account type can be seen here:

Tools|Account Settings|{select account}|Server Settings

If it's POP then the messages are all on your computer. So if you cancel the account, the messages are yours. The major irritation would be that when you look at the account, Thunderbird may try to contact the server and will show you annoying error messages.

If it's IMAP it is essential that you make copies, because the messages are stored on the server and you will no longer have access to them.

In both cases, copying the messages to Local Folders detaches and preserves the messages.

Right click the account in the Folder Pane on the left. Select Settings. It says there.