I accidently overloaded myself. Obviously, I can't run. Unlike Skyrim, in the Items menu there doesn't seem to be any way to drop anything. What am I not seeing? Help would be happily appreciated. Thanks.

Open your inventory with your Pipboy, place the cursor arrow on the item you want to drop and right click the mouse. When you exit the Pipboy the item will fall to the ground. If you don't play on a pc I don't know how you drop stuff, sorry.

360 Console is... scroll down to item and press X to drop.

Repairing items will also help with that as well.

In PS3 select the item and press square
You could also reapir said item with one of the same kind(but only weapons or armor)