Can you perform a "movement action" during your movement if the hero chose to use the Running Action? For example - Can the Hero having 8 movement instead of 4 now that he/she is running use up 2 movement points to drink a potion or re-equip? I believe the answer is yes, but would like to know for sure.

Yes, you can spend movement points when declaring a run action. So in your example, a Movement 4 hero declares a run, giving him 8 movement points. He can drink a potion (1 movement point cost), re-equip (free at the start of a turn, 2 movement points) and then move 5 spaces (8-1 for potion -2 for re-equip = 5 movement points to spend on moving).

You mentioned using 2 movement for potions, so I thought I'd make sure you'd seen the errata regarding potions: only one per turn now, no matter how much movement you have available. It definitely reigns in the craziness that can be achieved with multiple fatigue potions.