How Do You Stay Productive When You Are Distracted by Too Many Tasks

Context switching, practice it. When faced with multiple tasks, most people will try to resist, and turn everything off so they could focus on only one task. That does not work, if you are kept distracted. If instead you teach yourself to go from one task to another, with the goal of finishing the new tasks fast and quickly getting back to the original, then being distracted will no longer be a reason to worry. The distraction will no longer appear distracting.