Problems with my ex girlfriend and the whole situation

First of all, sorry for the long text. :)
To start from the beginning: we broke up, mainly because of mental health issues on her side, which also affected me a lot. It was more of a mutual decision because we had already started distancing ourselves from each other before that.
Then, about a week later, over the weekend, I found out that she was with another guy. The timeline didn’t really add up, considering we had only broken up about a week earlier. He said something like they had been together for about a month, WHICH WOULD BE PRETTY MESSED UP, but she said they only got together two days after we broke up. According to her, he had just been making moves on her for about a month before that.
You can probably imagine that I was pretty angry and hurt. We exchanged some messages, and I got a bit harsh as well, but anyway.
About another week later, she came over to pick up some of her things from my place. The atmosphere was pretty awkward and tense. But that same evening, she suddenly came back to my house and apologized for a lot of things—also for the way she had acted and, most importantly, for ending things with me, which she said was the biggest mistake she had ever made.
Up until that point, I had been trying to hate her, but deep down I always knew that I still loved her more than anything. So when that happened, it felt like a huge confirmation of my feelings. We even ended up kissing again in the heat of the moment.
After that, we still stayed in contact, and it was actually pretty positive. The problem is that she was still in a relationship at the time.
I talked to a few people about it, and everyone told me that I should just move on and stop talking to her. Honestly, I know that’s probably the best thing to do. But even now, I can’t forget her. We still have some contact, and I love her so much. I can’t even get to know other girls, even though I’ve tried.
It’s been about a month since the breakup now, and I know she’s done a lot of messed-up things. But deep down, I just want her back because I love her so much, even though none of my friends or family would support that decision.
The thing is, she can’t seem to let go of me either, which only makes everything even more complicated. But I just love her too much.
What do you think about this situation? Would you think it would be a bad idea if we gave it another shot?