The only FF which has kinda depressing story is FFX.
I don't find FF6 or FF7 depressing, they have sad moments but I don't feel depressed after I see "the end" while FFX... I love the game, I really do, but story is depressing, the ending is kinda hopeless, although it's a new start for Spira, but just think how many people on that planet died to Sin without any reason actually!
And if that wouldn't be enough (SPOILER)Tidus is a dream, no hope for Yuna, the most charismatic character Auron is dead, half of the Spira seems to be walking dead people! Seymours childhood is what made him to end how he ended, his mother sacrificied her life for him etc, etc. Do you really need more? I felt depressed for days after finishing FFX. Still it's a great story and great game nevertheless!