Damon wanted to hurt something - yeah, the kind of ache where you want to punch something into rubble. But the only thing that came to view was Red, but he knew he couldn't do it.
Present-day Claire descendants - both those dead and the countable few alive - call the excruciating pain they experience in the full moon the "werewolf virus" but in the past, it was known as "Elijah's heartbreak"
The pain they suffered on the full moon - the day Artemis watches over the earth in all her glory - was the amplified pain Elijah suffered being pierced in the heart by his father's sword. Artemis' prophecy came to fulfillment: the kingdom of Acadia won the war against the evil kings but the king killed his own son in disguise, blaming it on the kings just to have his son's bride. The pain presided and marked their suffering as Elizabeth wanted.