Well I'm sure if you ask around you can find another FC that will be nice enough to do runs without you rather than using DF. DF is cross server (but not cross data center. There are multiple data centers each with multiple servers)
While I do understand that as I came from WoW where healing is more ...healy, didn't you feel there were times when you didn't really need to heal and could maybe put in some other things?
Even in WoW, once a piece of content became super easy and we were basically in farm mode, I'd be asked to either switch to DPS spec (and let the other healer solo heal) or throw out a few things here and there while staying in healer spec. Maybe my group was unique in that way, and it certainly only happened once we reached "farm"-like conditions and not for any content where we were actually working hard, but it did happen. I definitely remember doing this on the first 2 bosses of the lich king raid many times.
Anyway, unless people are doing high-level content and you're trying and not just standing around (and not spamming heals when everyone is pretty much already topped off), it shouldn't matter if your dps is super high, most people will be content to just see you try. You can always try doing it gradually.
Comfort level -1:"I'm not trying" (hopefully you're not here): I'll heal when I feel like it but I'll do nothing even when no one needs healing for multiple seconds at a time and just watch videos or something.
Comfort level 0: I'm active but the only thing I'll do are healing spells and mana regeneration. Nothing else.
Comfort level 1: I'll heal whenever I feel I need to, but I'll at least keep one dot up as much as possible
Comfort level 2: I'll heal when people get < 70%, keep one dot up, and throw attacks out when above 70%
Comfort level 3: I'll heal when people get around 50%, but other wise keep learning to dps. If I'm playing a class/job with more than one dot, I'll try to make sure they're both up and also throw occasional attacks.
Comfort level 4: I'll see how far I can push it while not being overly stressed out.
Comfort level 5: END GAME: I'll see how far I can push it as long as we don't die even if it stresses me out
These comfort levels can change for each instance, and depending on how good the rest of the group is, but the idea is to try to work towards the highest level possible.
I would also recommend finding someone in NN or FC to practice casting heals on to get a feel of how strong your healing skills are so you can best determine if it's necessary to heal or if you can wait a bit and throw out another Stone.