I currently have brown rot, and I can't move at the moment because I'm overencumbered. I typed in player->addspell cure common disease, and I get nothing.

And is there a console command to make NPCS unequip or equip things?

Diseases are technically just spells, so you want to remove the spell.
Player->removespell "brown rot"
or just give yourself a Cure disease potion and drink it.
player->additem "p_cure_common_s" 1


You can remove items from NPCs the same way you do it for your character.
"ID of the NPC"->removeitem "ID of the item" 1

To force them to equip something you first have to add the item to them (same way you would remove it) then use:
"ID of the NPC"->Equip "ID of item"
(this function does require that Tribunal is installed)