Yes, you are correct in assuming that. For the achievement, you need to complete every quest within one playthrough.
Killing Crawmerax in PT1 should be easier than doing those Circle of Duty quests, cause after killing Crawmerax in PT2, all enemies in PT2 should have scaled to his level.
You should try out the Crawmerax Glitch spot:
You may have to read through the thread a bit, though.