I originally had an authenticator set to this character, but since adding it to my jagex account, I've never once had to enter that authenticator code. Do I even need the individual authenticator codes for each character on a jagex account anymore?
No, Jagex accounts have their own authenticators that would be tied to them if you choose to use it instead of the email auth.
Character auths can be discarded since you cannot use the old character's login details anymore.
Can I turn in a paper without citing all sources?
"No."[1]
1. William Shakespeare,
Hamlet
, Act III, Scene 1, line 96.