The first puzzle involved 4 levers, but I accessed the barrow before I got the quest. Since i was impatient and didn't want to redo the puzzle I simply flip a switch that closed the door to the levers but opened the exit door while simultaneously leaving the room. Upon re-entering the barrow I learned that I couldn't open the door to the lever room so I couldn't progress any further. I could revert to a previous save to fix this problem.
(Well aware this thread is a little old) If you flipped the lever and exited at the same time, then you can't really access it again, and you'll have to reload. However, due to this thread not being re-replied to, I assume you've solved the problem.
Re-entering Shroud Hearth Barrow a day or two later after finishing the quest "Wilhelm's Specter", I encountered a locked door on the upper level south of the oil pit chamber. Trying to open it produces a message that it requires a key. Where is this key supposed to be found? I think that I picked the lock when I went through this dungeon originally.
BTW, it seems to be a characteristic of all dungeons named "...Barrow" that they require a claw. Can anyone verify?
I was still able to return to most of the dungeon to collect more loot, because the short cut from the word wall chamber in the "depths" section back to Wyndelius's Study was still open