How important is your private oboe teacher?
VERY.
When I get students who have studied primarily from lesson books (Rubank, Gekeler and the like) I nearly always have to correct fingerings. Those books have been out for eons and they’ve never been corrected! They don’t know about left F, for one thing. But there are other issues as well. In addition, some band directors continue to insist on using the E flat key when a student plays forked F.
Get a private teacher. And get one who is actually an oboist!
Trust me on this.