EDIT: Sorry, missed that the OP is an SCH, not a WHM. Fixed class-specific commentary/advice.
As someone who often tanks dungeons, overhealing in dungeons is terrible and stressful. Please don't do it. If a heal spell will heal someone to max in a dungeon, hold off on casting it until that's not true. Of course, AoE heals are fine if three people need it even if the fourth doesn't, but other than that, overhealing just puts stress on the tank. Overhealing is the quickest way to gain aggro in the game*. This is especially true in dungeons before level 40 if your tank is a PLD and dungeons before level 30 if your tank is a DRK or WAR. That's when each class gets their tanking stance.
While you don't have to DPS as a healer, in many dungeons, you'll find yourself mostly sitting around doing nothing if you aren't DPSing. Especially with your fairy healing. In fact, you can do most early dungeons without ever casting a single healing spell and just let the fairy heal it all. If you are just following people around and not using any skills, it might upset some people, especially since it makes you look AFK or like you're botting.
The dungeons will also take longer. If you math it out assuming healers and tanks both deal 2/3rds the DPS of a DPS (the actual number should be much higher), your party will do 20% less DPS without you, causing the dungeon to take 25% longer than it would otherwise. So, a 24 minute dungeon will now take 30 minutes. That much of an increase will upset some players, though others won't care.
But, if you have zero interest in doing damage ever, period, you are playing the wrong game. How the heck do you solo as any class without doing damage? Do you have someone else always come along with you so they can kill everything you encounter so you don't have to DPS? Even if you do that, there's the solo missions, and while there are sometimes NPCs to heal, most of the time you HAVE to do damage to succeed. I can't think of any MMO where a healer never needs to attack...
If you are okay with doing damage, but are worried that if you do it in dungeons the healing will be too hard, then ease yourself into it. At first, never use Cleric Stance and only use Bio I. When no one needs healing, go DoT some creatures that are not currently under the Bio I effect from you. Once you get used to that, start throwing in a Miasma (or Ruin I/II if all monsters are Miasmaed) whenever everyone is at over 90% health and all the enemies are DoTed already with Bio I. Later on, you'll want to use Bio II on all the mobs before resorting to Ruin I/II. Once you get the hang of that, start using Cleric Stance before casting Bio I, but stop using the other DPS skills for now. Get used to turning it on, casting Bio I on everything you have time for, and turning it off. Cleric Stance doesn't take a GCD to turn it on OR off, but you can't turn it off immediately after turning it on (there's a cooldown). You CAN turn it on immediately after turning it off though, so be careful not to doubletap the button. Don't go casting Miasma/Bio II/Ruin until you get used to "stance dancing" with Bio I. If a dungeon is proving difficult or hard, then don't DPS at all, just focus on healing.
Also, in the endgame, there are many dungeons where people will go fast enough that you won't be able to DPS much. Many of the later dungeons, the tank will gather 2-4 groups at once and you'll be hardpressed to heal fast enough to keep them up. That method of play is pretty stressful on the healer, but it seems like the sort of play you're going for? It's far harder and more stressful than just DPSing during a normal dungeon run though...
* Case in point: In this game's endgame, there's a Hunt mechanic, where you find a certain type of creature in the open world, get a group together, and kill it. To get full credit for the kill, your party has to generate a certain amount of aggro on the monster. The best way to ensure full credit has always been to have a healer or two do nothing but overheal. It generates about 2-3x of what a good tank can generate in the same time.