Is it possible for a character to perform running shot the same turn that they broke away from an adjacent character?

Decalre the action. Roll to break away. If you succeed continue on with the action. If you fail then your action is over.

In other words... yes, it can be done.

Just to add one thing:

If you are using Warbird and want to breakaway from some mook and use Running Shot to attack someone else you have to declare you are using RS when breaking away.

The reason is this - if you succeed you can move Warbird up to 5 spaces and then make a ranged combat attack. If you fail, however, you cannot use the Avengers Free Move TA to not have the failed breakaway cost you an action.

Running Shot is a Power Action while the Avengers Team Ability allows you a Move Action that doesn't cost you an action from your total.

I hope that makes sense...