I'm so confused (transfer of characters from TBC Classic)

Is your old realm in the TBC realm list still? Recently a lot of realms were closed down and all characters were moved to Mograine. If you are 100% sure you chose TBC try looking there.

When Classic TBC started, before you could play you had to choose Classic Era or TBC, you had to pay to play on both. Would you mind using the Terms TBC and Era? You said you logged in after TBC, do you remember which you chose for your mage Era or TBC?

When you log onto Era, is your character there, I mean can you actually play it? If yes, then unfortunately it’s stuck there. If it’s there as what I call a ghost character, ie you can see it but not play it, then your mage may be on Mograine.

Edit: Having looked here.

I can see your old realm was shut down, so if you had any TBC characters there they were moved to Mograine. You may need to either make a new character there to refresh the list or if that does not work, open a ticket to ask about where your mage is.