Excellent advice!awholley said:The "correct" answer, though will be more a matter of voicing -shaping your oral cavity, tongue and embouchure to facilitate the changes...
My suggestion is to stop thinking of slurring down to G from E, but rather to try thinking of and voicing for low G while you are playing the E. The octave key will make the E come out anyway.
The initial exercise to facilitate this would be to practice slurring UP (and back down) from low G to E while leaving the G embouchure/voicing intact.