This obviously doesnt apply to just Falcon and Captain America. To use Falcon's attack boosting power with running shot or charge, do I need to be adjacent to Falcon BEFORE I move or AFTER I move?

after you move

The modifier is activated when the attack is made. So depending on the circumstances, the specific timing may vary, but generally Cap and/or Falcon would be attacking after the movement and that is when they would need to be adjacent.

If, for some reason, they attack were to occur before or mid-movement, well, that would also be the same time they need to be adjacent.