Are the two locations given for each ring randomly generated? or are they always at the same two spots?

I finally got this quest only to discover that I've already cleared out one of the dungeons quite a while ago. I violated a Cardnal Skyrim Rule: Never randomly enter and clear dungeons without a quest....which is what I did here .
The white arrow pointing to where the Ring of the Beast should be is just floating in the air above the trap door in Haldir's Cairne...only, again, I've already killed him. His staff decorated my wall in Breezehome before Hearthfire

Right now I can go back a few saves with no issue and continue with other radiant quests as they come, but sooner or later the rings will come up again. If the Ring of the Beast will always be in Haldir's Cairne then I'm cooked.