Assuming your healers aren't asleep at the wheel, sounds like you are likely dying from avoidable damage. If that's the case, you should try to focus less on pushing out as much DPS as possible and focus more on the fundamentals of avoiding damage.

Avoid standing in AoE markers. Even if it means cutting a cast short, cut the cast short. The biggest part of learning Black Mage is knowing fights so you can avoid as much damage as possible while still being able to cast. And remember, cutting a cast off early is a small DPS loss but being dead is a big dps loss. So try to focus on what's happening around you. Also as a random thought which is probably pointless since you are a caster and not phys, don't stand in front of bosses. They often have untelegraphed cleaves that a Tank survive with cooldowns, but DPS and Healers will get shredded by.