That no one commented on the shamans top healing being done by his/her 3 minute cooldown tranq-totem on most attempts is beyond me. Yes, the healing is bad.
No, that's not why you're dying. As said, mass dispell is a must. Your shadow priests' can't go OOM from using it every 15 seconds as long as they keep vampiric touch up on 3-4 targets, and they can always use dispersion to regen mana if they really have to. Tranq from a shadow priest doesn't heal for anything, so you might aswell not use it. You should still have plenty of cooldowns to use against rain of blades. (druid tranq, shaman tranq, rallying cry, revival, your own HOTW-tranq).