Yes, you need to finish Sastasha to advance. And a lot of the story contents such as your personal chocobo and grand companies don't open up until you are further along.

As for the duty finder, it's very difficult to get matched right now as a DPS since there is an acute shortage of tanks and healers. The algorithm prioritizes near-complete teams first so if you have 2 DPS and a tank already matched, they will likely get the next healer that signs on, for example. DPSs only get fast matches if there is an existing group that lacks one DPS (unlikely since there are so many), otherwise they will need to wait until they get lucky and an unneeded tank gets matched with them.

A good way to reduce wait times is to create a party with at least one other player. Starting out with a two person party in the duty finder will place your far ahead of most others and makes the system prioritizes your party when assigning matches to you since it favors more complete teams than individuals.