Hi
Gearing up as a healer is fast if you know where to go
Do the Crucible of Carnage in Twilight Highlands - nice weapon upgrade if you cannot buy one from the AH
Try to get your item level as high as you can by purchasing epics from the AH and replace what you can with Justice Points that you have earned if you completed dungeons while leveling.
You should be able to enter the troll dungeons and just run those until you can replace the lower items.
You will be at a higher item level for Bastion of Twilight, Blackwing Descent and Throne of the Four Wind raids - Im not sure if people still raid these instances as they quite old now.
Get to a high enough item level so that you can do the 3 dungeons that were released when Dragon Soul came out, those have quests for completing the dungeons all with good gear.
You can try find some Firelands raids now, On my server they are still running them quite often.
From there you can start raid finder and get loot from there.
If you struggling to heal on the heroic dungeons, do the normal versions until you get enough Justice Points to replace your gear from leveling.
Also note that all epic items bought with valor points have changed to Justice Points, so dont go and buy an ilvl346 item when you can get an ilvl 397 for the same amount of points.
This did take me around two weeks or so to get good loot, it does take time so don’t think it will happen overnight unless you know people in guilds that have Dragon Soul heroic on farm and can boost you

IMO the troll heroics are really great for stacking up Justice Points (if you cannot do the Dragon Soul Dungeons yet) mainly because you don’t have to clear all bosses to get the completion.
I should mention that the Dragon Soul Dungeons are the following:
Well of Eternity
End-Time
Hour of Twilight
Item Level Drops:
Heroic Dungeons - 346
Bastion of Twilight etc - 359 & 372(heroic)
Troll Heroic Dungeons - 353
Firelands - 378 & 391(heroic)
Dragon Soul Dungeons - 378
Raid Finder Dragon Soul - 384
Dragon Soul Raid - 397 & 410(heroic)
All I can think of at the top of my head, hope it helps