OK, now that I've read about these Flip-Flops, let's see if I get this right:
The J-K FF acting as a T-FF:
J&K will be pulled high all of the time (mimicing the T), and the Push Button goes to Clock, which is normally LOW (since the button is normally open). Pushing the button sends Clock HIGH, and then releasing the button drops it to LOW again. Because J&K are HIGH, as the Clock dropped, the output on Q and /Q are both toggled to their complementary. I would connect Q to Output-A and /Q to Output-B from my original diagram to turn one off and the on at the same time.
As for the D-FF:
Tying /Q to D acts as the clock for the FF, creating the rising/falling edges needed to capture data. In this case, the input on D doesn't matter, and I would use S&R to trigger the different state. Not sure how I would set both S&R to opposite states with only one pushbutton?
In either case, how do I keep 12V flowing to Output-A and Output-B after the toggle event?
Here's the full circuit diagram. Anything in BLUE is my wishlist to the circuit. The black is the original circuit.
The original circuit is normally used with a toggle switch (where my OUTPUT A & B are), that activates a twin-coil switch motor. It dumps a capacitor charge to one side while simulataneously starting to charge the other side, in anticipating of the next toggle/dump. That charge & hold process is why I need the constant 12v to the ON side of the trigger. It also controls the LEDs display that show which side of the switch was last thrown. Hopefully this helps put it in perspective.