How about making Draven's Q cost more mana to use?
45 mana is not really much given how much damage the buff allows you to deal. Especially in the early game when a 1 or 2 component item advantage could chunk the opposing ADC by a third of their max health. I get that he has to catch the axe to keep it going, but dropping it only costs him 45 mana, which if we are honest, is not very much of a cost for chunking the enemy ADC. By the time the CD is up, he has regained most of said mana cost from just passive generation.
So why not make the guy work a bit harder to keep his power where simply slacking off on catching an axe is no longer just an annoyance for a few seconds while it comes off of CD.
Maybe make his Q cost something like 100 mana or so? That way he cant just ignore the axe catching (which are already not too hard to catch anyways) if it is in a mildly inconvenient spot. That being said, I do not want to gut his early game completely. So since he would no be able to spam his Q without much of a risk, due to the higher mana costs, how about every time he catches 2 or 3 axes in a row, he gains 5 or so mana back (or something like that)? It would seem like a fair trade off.