Call the DE State Police and ask, or look on the DE.gov website, they have the state code and firearms statutes that will give you state-specific information.
I'm fairly certain DE has no restrictions on FMJ. In a .380 semi-auto, I would carry FMJ, with a heavy-for-caliber projectile, and one reload. Any shots taken will likely be at very close range, and accuracy beyond 7 yards is unimportant.
That said, I think you'd be better off with a Glock 23 (.40S&W) or Glock 30 (.45ACP). And some training, both general, and specific to carrying a trigger-safety firearm.