breaking away is considered a move action correct

and running or charge is a power action correct

so the 2 cannot be used together...correct

Wrong. Anytime you try to move when in base to base contact, you have to perform a breakaway - however, breakaway is not an action in itself but rather an effect of moving. You can roll breakaway in order to do a Charge or Running Shot.
However, you have to declare why you are breaking away prior to rolling breakaway - if you want to Charge with Wonder Woman and fail breakaway, that's still a power action and not a move action (and JLA TA does not apply...)

Note that this is a change from earlier rulebooks. It used to be that a failed breakaway somehow changed any action to a Move Action, thus the Avangers/JLA/ect. TAs would apply.

Fortunately, that has finally changed....