The first time I played this I solved it on the first try, but now only the lever with the eagle works. All the others just keep resetting themselves when I pull them.

It seems the puzzle doesn't work after you solve it once. Did you already visit the ruins and kill Vokun? Try the console command tcl (toggle collision detection) and walk through the floor grate.

No, the second time was after I had started a new game. I'll try your "tcl" command and see if it works.