I'll start by saying that I don't like running FATEs, especially not in big groups. However - leveling my THM I've only just hit level 16:

I have no quests to do.
I have completed 3 level 18 and 1 level 19 quests which I can't turn in.
I have been queued in the duty finder for about an hour, including having the "join party in progress" option checked.
I have cleared the second tier hunting log but haven't unlocked the third tier.
I am out of allowances but even when I get them I'll switch to my main to do the grand company quests for seals which I can't get on my level 16 character.

The only things left (while waiting for my queue to pop) are:

FATEs
Grinding non-FATE mobs

Honestly I'd feel a little guilty doing dungeons on a dps alt when so many people are having a hard time getting them done for their story quests even if I did get in (especially since I'm already geared from greed rolls running on my CNJ).