Ugh! Why not create two distinct addresses?

I think you could add the second name as a secondary Identity of the first. In account settings, look at the "Account settings" panel and click the "Manage Identities" button at bottom right.

You'll have to use the same address (since you only have the one) but you can set the "display name" that shows alongside it.

Changed the Display Name in Settings on computer B to computer B's name. This solved the From name, but not the name in the Sent Folder. Goes out as from B's identity but shows up as Sent from A's identity in the Sent Folder. Strange, as A's identity is nowhere in B's settings. Thanks

Is this gmail? It silently replaces the "from" address with the one associated with the selected gmail sender. If B uses A's smtp server, B's messages will appear to have come from A, but your Sent folder may tell a different story.

No, this is not gmail, it is cox mail. Computer B was using Windows Live Mail until the Creators Update caused it to crash and windows had to be reinstalled. Since Live Mail is no longer available, switched to Thunderbird with the same cox mail account used before. Both computers worked fine on Live Mail in the past, then with computer A on Thunderbird Mail and computer B on Live Mail, but now with both on Thunderbird having this problem. Same email account the entire time. Thank you for trying to help me.

if you add the email address to the address book, that display name will appear in Thunderbird folders. So if Mr Bruce email Mrs Bruce, the email will come in with His display name. When Mrs Bruce replies the email address and associated display name will be added to the addressbook as Mr Bruce and all mail in all folders will use that display name.

There is an option in options to disable the use of the display name. But like Zenos, I suggest you just use two email addresses.

The "From" name was solved when I changed the display name in Account Settings. The "Sent" Folders still show the same name on both computers. Trying to change the name on the "Sent" Folder so it shows the same name as the "Display" name on each computer. Thank you.

TBird18 said

The "From" name was solved when I changed the display name in Account Settings. The "Sent" Folders still show the same name on both computers. Trying to change the name on the "Sent" Folder so it shows the same name as the "Display" name on each computer. Thank you.

Options > Display > Advanced and deselect the show display name option.

Thank you, Matt, that worked!

Options > Display > Advanced and deselect the show display name option.