Do you have "The Swallow's Compass" unlocked in your Duty Finder, it moved from the right dungeon tab to the left dungeon tab.
the right dungeon tab is always just for the dungeons of the most current expansion, so from level 71 onwards. Every other dungeon is listed in the left dungeon tab.
The same goes for the trials tab.

If you completed "Tortoise in Time" your next quest in this quest line is "The fire-bird down below" and after that "Surpassing the samurai" all will be offered to you by Bunchin in Tamamizu.