Struggling w/ sudden death and blaming myself

Yesterday, my fiancé and I were forced to say goodbye to our cat at an emergency clinic at 5AM. It was very sudden, and I'm really struggling with it. I've said goodbye to cats before, but the previous ones had been very old and had full, long lives. This is the first time I've had a pet die from illness when I still felt like they had a lot of years left, and I'm struggling to not blame myself.

For context, my fiancé and I are moving in together at the end of the month, and while I've loved this cat for like 6 years, she had only moved into my place a little less than a month ago. My fiancé noticed the cat had been acting sick and losing weight, so we took her to the vet and went ahead and moved her to my place because that is where we slept most nights anyway, so it would be easier to give her medicine. The vet basically brushed it off as just being a cold and told us she would heal up on her own, so my fiancé had to explicitly ask that they prescribe some kind of medication that we administered over 3 weeks. Well, we administered the antibiotics, and she had started to have a little more energy and was eating/drinking more, so things looked good.

This past weekend, she was very lethargic and went back to not eating much, so we scheduled a follow-up appointment for the vet Tuesday, which was the earliest they could see us. Monday morning around 4AM, she woke us up with kitty death howls while she was tucked away behind our toilet. We rushed her to an emergency clinic, and they told us that she was yellow probably from either "anemia or liver failure". They came out 5 minutes later to tell us that she had passed.

I am really messed up by all this. I have loved that cat for a long time and was so excited to have her at my house and be a lil family with her and my fiancé. I got maybe just under a month of her being at the house and snuggling us in the bed while we slept. I had always assumed I'd get to experience that for much longer, but now that's just suddenly gone before we even realized she was seriously ill. And I don't understand how she was either anemic or experiencing liver failure when the vet basically brushed it off as a little kitty cold. I am hating and blaming myself for believing his optimism and not doing more to ensure the cat's health. I am worried my fiancé is going to resent me for it. And mostly, I miss Lady (the cat) and am absolutely heartbroken that I got a one month tease of being a happy little family with her for it to suddenly be ripped away.