One thing that might help: Don't go for the fifth orb in the second transition. Instead, use the time the 5th orb travels to dps the boss. How much HP does he have when the last phase starts? It shouldn't be more than 35%.
As for why people die in the last phase, I would say your healers suck. During the last phase they should be able to do 300k HPS together (including CDs), which they are not delivering. I don't know much about Paladins, but the Druid could definitely do better. TBH I don't know what he's doing in the last phase, he should be using the heck out of Rejuv, but only had 40 ticks (I had 171 in the same time). Does he have mana problems? You healers should enter the last phase with near 100% mana, or they're doing something wrong.
Compare the healing in the last minute of your best try with the last minute of our kill yesterday. We used Healing Tide during the Transition, and Tranquility during Bloodlust. I did 170k HPS in that minute, our off-spec healing Shaman did 125k. Your healers need to step up their performance. And yes, don't use Priest Tranq, it's a waste of dps. Your shaman should use Healing Tide / Ascendance though.