Running and shooting lol, sounds op, hell walking and shooting sounds op. But it would be awesome if achievable, and like you said give incentive to AP regen+ which I feel is currently lacking.
Perhaps only allow hex-shooting while running with light guns, and walking will allow hex-shooting with big guns (not walking while deployed though).
Aside from the creation of an entirely new mechanic (and all the headaches bound to arise), a damage buff wouldn't go astray here IMO, melee is still capable of dealing 200-400 damage in a SINGLE hit (more on unarmored targets), unless you planned to put the damage into those regions I think melee would still be viable.