One option you might consider is to add either a tank or healer class yourself. Even if you want Dragoon (or any other damage dealer) as your primary job you play through most of the storyline with, if you keep multiple classes/jobs up to the level you're at in the storyline, you can alternate to your tank or healer when the story hits one of these dungeon/trial instances and get a fast queue.

You'd still want to run dungeons a times with your primary DD job as well (for XP, to practice how that class/job works in a group, to be able to "Need" on role-appropriate loot, etc), but if that's just in Roulette and when you have plenty of time for a long queue, you'd be less trapped into needing to wait on a long queue at an inconvenient time before you can continue the story.