Catching while chasing shouldn't be too hard as the axes go in the direction you are traveling right after you throw your axe. However, team-fights are always going to be rough for Draven. Try and aim your axes to land in a relatively safe spot that you don't think is in CC or stun range. I always walk sideways after an attack, that way the axe is closer to me and is in the same relative range. Walking backwards greatly delays your attack speed and standing still is suicide. If you have to though, don't be afraid to walk backwards.
Honestly, Draven's damage doesn't' fall off too much without his axes - he'll be about on par with most ADCs if you don't use them. So think of using axes for the first two-three hits and then focusing on positioning and chasing. If you get those first two or three axe-hits your damage will greatly outscale any other ADC. Then just finish off with auto attacks.
Also, practice with axes as much as possible - you have a whole laning phase to just play around with your mini-game. Take advantage of that. Sometimes I juggle two axes in lane phase just to get better at it.