What comes to mind to save loads of Hbridges is a 16v centre tapped supply ( with big electrolytics to supply the pulses )
with the dots coil connected to the 8v centre tap.
A transistor ( or ULN2003 for 8 dots ) would ground the other side of the coil for driving one way, and a high side version of the ULN ( I cant remember its number ) would pull it up to 16v to drive the other way.
As with H bridges, you would have to ensure both drive logics are not on at the same time...