Only the DRG questline so far heavily impacts the MSQ (though Estinien). There are cameos and references in other ones but it isn't exactly 'breaking' to do them out of sync with the MSQ with the 30-50 MNK being the exception. That said, a lot of the jobs get at least references from time to time.

As for Jobs that follow a story close to the MSQ, MNK is an obvious one. It deals with a pretty significant aspect of Ala Mhigan society and history. NIN and SAM are thematically relevant but their actual stories aren't particularly tied to the MSQ. DRK is but only in the sense that it addresses an aspect of the WoL that isn't really addressed otherwise and that being how the WoL emotionally responds to everything the have been through. Honestly I think they did a stand up job of making it personal while keeping it vague enough that any normal mentally sane person would probably have similar feeling on some scale. DRK harness feelings of pain, loss, anger, frustration and fear and pump Aether into those emotions to manifest the dark fire they use as a weapon. As such the MSQ touches on these in relation to the WoL and their experiences in the MSQ. If your character has never felt any of these emotions then DRK wouldn't be a Job that really makes sense.