Hi OP, I just wanted to say, having been the child in this situation, it's so refreshing to know that (1) this has an actual name now and (2) people can get help for it.
Once I reached adulthood, I always characterised my situation as a 'Munchausen-by-proxy kind of thing' or 'yeah, my mum's a bit Munchauseny like that' but I knew it wasn't quite that - health anxiety by proxy is exactly what it was!
As an adult I have learned more about DM and understand that what I saw as extremely controlling behaviour growing up actually comes from a place of huge anxiety for her. And not just health-wise - she came across as completely unhinged when I started going out as a teenager (listening to my phone calls, confiscating my car keys, following me on dates, etc). In her head, it was all to try and keep me safe and she was genuinely terrified for me, but her anxiety went completely unchecked and she lost all perspective.
Our GP noted it on my medical records when I was a child (IIRC he described her as 'overbearing'), but didn't intervene or say anything to her (I mean, drs were sharp with her and admonished her, but didn't actually suggest she might have an issue or actually help either of us - it was a different time).
Medical appointments were always really stressful (and often quite hostile) - I used to dread them, even when I started going on my own as I felt like the boy who cried wolf by that point. I often just didn't bother going, and even now I find it hard to advocate for my own health. Medical professionals ask 'why did you leave this so long?' 'You've been living like this for ten years and never thought to go to your GP?' I make some excuse, but I know why.
I can't tell you how great it is that you have the self-awareness to recognise your anxiety and have had the presence of mind to do something about it; it's the absolute best thing you can do for your DD and she's incredibly lucky to have you!
It seems like it's still early days for the Sertraline to kick in, so give it time. Have you tried meditation? Theres an app I really like called Simple Habit - it has normal relaxation things, but also mini-CBT courses for all sorts of topics: anxiety, parenting, stress, etc. I love it and my DD also finds it very helpful for her severe anxiety (circle of life!).
One thing I think that might also help around your DD is a bit of 'fake it till you make it', trying to come across as a bit more chilled around her, even if you're not feeling it inside. Not checking her over and consulting books (or I guess the internet nowadays) when she's around will help her, as I know my DM doing the opposite caused me a lot of stress and anxiety!
Good luck - what you're doing is so amazing for your DD and you'll both be fine! 