I often have problems going extremely quickly from the upper register to lower register. For example from E in upper register to E flat in the lower register. I don't know why this happens. Maybe I tongue too hard?

i find this problem aswel.
i find this is most commonly one of 3 problems
1)in the higher register, you tend to have a tighter embouchure, and this makes a really sharp note in a lower octave, a squeek, or even playing as if with the register key on
2)sometimes my thumb lingres on the register key, so i can drop to the note, sometimes its a combination of both
3)tounging the note before finishing moving the fingers

focus on your embouchure~ be sure to relax more on the lower notes, the fingerings isnt as necessary, but its annoying if this is the problem
hope ive helped~good luck