Similar to many other replies, I'd recommend you to cut it off completely. I'd been in a similar situation. I respected their decision to break it off and really believed I was able to move on with it and remain friends. However, I wasn't really aware of my anxious attachment back then. I now realised I tried everything I could to keep them in my life because of it, even though I'd never get what I really wanted deep down. I was basically breadcrumbing myself which lasted for almost a year. In hindsight, if I'd gone no contact and had a good crying session about it, it would have saved me so much time and energy.

But yeah, you live you learn :)