Remember the days when we didn't have gear score or that little paper doll telling us what our average iLevel was? Remember pouring over guides and loot lists to see what numbers we should be hitting, like Spell power and Crit/Haste? Far too often, I see people fall into the trap of "Well, my ilevel says I can get into a heroic, so it must be true." I did the same thing and started to wonder if there was a flaw in the system.
Is there a flaw?
Yes and no.
Yes, because people are by passing requirements by just having gear in their bags that are higher level. Then, it's no wonder there are cries of heroics being too hard or that it's punishment for healers.
No, because the system is just crunching numbers. It is not designed to tell you if your spec is appropriate or that you are wearing enough gear to be viable. It's just a number. An average number, at that. It is not supposed to replace human thought and comprehension.
So, my recommendation is this: treat the score as it is- an average of all the crap in your inventory. Then, remove it from your focus.
If you are having a difficult time adjusting to the dungeons, heroic or normal does not matter, take some time to really look at the quality of gear that you have. Is your gear 333 or higher blues or are you plugging along at 312 greens? Are you wearing stuff with hit on it as a resto? Have to checked to see what reputation rewards are out there for you? Heck, even quest rewards can be substantially better than a drop.
Perhaps we need to go back to some basics. Work on those 85 dungeons, grab the 333 gear, rep grind some. If you have access to them, find some gems and enchants (which, depending on your guild and cash flow, may be difficult to do. I'm still using a Wrath meta gem because JC sucks to level).
Of course, this does not factor in what actually happens while in the group (usual stuff like teamwork and strat basics), or even you personally are capable of. But I can tell you that my first heroic in the minimum required gear still left me feeling like I had no right to be there. I was in greens and low blues, with a few pieces to get just above the random mark. I told myself that I wasn't going back to a heroic until I made some changes.
Your experiences may differ but I firmly believe that the number does not tell the whole story and that it is up to the player to fill in the blanks.