If you want to learn SGE, then I recommend running dungeons. Bozja is great as an exp source but it teaches you very little about playing the class unfortunately, same with fates. Doesn't mean you shouldn't do them but for practice, dungeons are still miles ahead of everything else.
If you can grab some friends, ideally one on tank, that would make it easier and smoother for you until you feel more comfortable but just writing at the start that you just picked up SGE and don't feel confident on the class yet helps. If you fear you won't get it typed in time, make a macro.

I don't remember too much about the SGE quest but I definitely don't remember having any difficulties at all there.
- is your gear up to date? I think unlocking SGE only gives an okay-ish left side but no right side.
- what are you healing her with? Diagnosis is incredibly weak, even Eukrasian Diagnosis is generally fairly bad. Do you use Taurochole, Druochole, Haima and such? Iirc aoe heals like Kerachole and Physis II also heal NPCs now but even without them, Haima, Tauro, Druo and Soteria is plenty with the auto heals from Kardia.
You may still need to the occasional EDiag if you run out of Addersgall stacks but healing really shouldn't be the problem here.