Can I use Lingering Souls' flashback ability the same turn I cast it normally?
If the card is in the graveyard, you have the mana to pay the flashback cost, and  priority to play the spell, you can play it. Note that you have to wait for the spell to resolve before you can play it again, because that's when you put the card in the graveyard.

There is no rule specifying that you have to wait until your next turn.