Barnes: Playing the (high) role of Prince Keleseth. Your win rate when you play Barnes on curve goes up dramatically; feel free to just slam him as soon as possible. If you pull him later in the game, consider which minions are still left in your deck and try to play him when one of those minions would be most advantageous to you.
Eternal Servitude: The strongest card in the deck once you’ve gotten a minion down. Just be careful if you’ve used Potion of Madness liberally; the resurrect pool may have fewer usable options than you’d like. That said, I don’t think I’ve ever played Servitude and not had a usable pick.
Greater Healing Potion: Sometimes you just need to heal for 12.
Mass Dispel: With all the cards that provide buffs to minions (Murlocs, Keleseth, Cobalt Scalebane, Mark of the Lotus, etc.), being able to reset a board that is getting out of control is valuable.
Shadow Word: Horror: One of the most underrated cards in Priest up until this expansion. This usually gets fine value on its own, and paired with Pint Sized Potion is your only clean answer to 4-attack minions outside of lucky deathrattles from Obsidian Statue.
Dragonfire Potion: Your most premium removal. You have lots of options and two copies of this card, so you can be a little less stingy with this than when playing Highlander Priest, but still try to save it for boards where it will make the biggest impact. Also remember that the “except dragons” part is occasionally relevant, most often when you have Ysera or your opponent has Cobalt Scalebane.
Shadow Essence: One of the obvious drivers of the deck, because it cheats out a big minion that you can resurrect. One key misplay many people make with this deck is going all in on this card when they shouldn’t. If Barnes hasn’t been drawn, you’re effectively playing Russian Roulette when you play Essence, which is fine if you know the percentages; every minion you’ve drawn increases the odds that the pull will be Barnes dramatically. With all your minions in the deck, the odds are 17% that you’ll pull Barnes; if you’ve drawn 3, it goes up to 33%, and with 4 drawn it’s a coin flip. When you’re behind on board on turn 6, understanding how likely you are to get a minion with taunt as opposed to Barnes, especially if you have other removal options to choose from, is key to success with this deck. Don’t just blindly go into Essence on 6 every time because it’s not always correct.