Since OP has a soul crystal as a level 90 Dragoon, I doubt that is the issue. Jeeqbit's solution works nicely, the recommended solution as a physical DPS is to work on each sword one at a time, ignoring AoE attacks and using DoTs where applicable. [That also worked great as a Bard who could move].

Also, the quest is much easier to resolve if you use the 'easy' button when entering after a failed attempt. Less damage to you and more damage from you.

The iLevel 400 weapon (Allagan Tomestone of Poetics in Rhalgr's Reach [still 600 Poetics?]) will help with the damage. If you have a full set of Augmented Scaevan armor and/or accessories (available in Kugane or Rhalgr's Reach for Poetics, which are easy to get if you do roulettes a bunch) it'll also help.