First thing I would suggest is evaluate why people are dying. It's good to be willing to accept responsibility for mistakes but there's a saying amongst career healers; you can't heal stupid.
If your tank is poorly geared, not using CDs properly, standing in AoE especially during large pulls or doing any combination of these it can make it almost impossible to keep them alive, especially once you exhaust your oGCDs and buffs.
If your DPS are standing in AoE or ripping hate off the tank that's also not on you - of course you should try to heal them but sometimes it isn't possible. I was doing a few latelatelate night Byakko runs with some friends the night before last and the BRD has her BiS - she died several times to auto attacks because she just couldn't dump her hate. That was 100% not the healers fault.
If you find that most of your situations occur on large pulls as opposed to bosses you cab also consider the WHM-specific stratagey of Holy for mitigation. You can stunlock enemies for almost 2 GCDs with one Holy cast and you can Swiftcast it to make it easy on yourself. This can be nice to get a regen on the tank while he isn't taking damage and toss down Asylum. It's also great damage as I'm sure you know.
Don't get discouraged - as Sebazy said leveling dungeons can be some of the hardest dungeons you complete. Also do your part in minimizing your party's ilvl disparity - walking into Doma Castle in Shire gear isn't making it easy on yourself. I would update the majority of your gear around 66, most Market Boards have HQ 66 gear for pretty cheap overall and it can help suppliment things you haven't gotten dungeon drops for.
Lastly you're on Aether - if you want someone to run some of this with I could always tag along on RDM as both a safety net for quick raises and to see if I can spot any holes in your WHM-ery. I'm no Sebazy but when it comes to dungeons I'm a pretty good WHM. Discord is moromurasaki#5719 if you wanna say hi.