You two were in a relationship, so you are used to doing things together. So, even though she broke up with you, she still craves that time and companionship. And since you are the one she has had that with, she is still trying to have it. It will stop once she starts getting involved with someone else.
The fact is that she broke up with you, so I wouldn't even think about a reconciliation at this point. Whatever the reason for the breakup, that has to be completely resolved before you can even think about getting back together.
But if you do not want to be friends with her, then you need to tell her that. Tell her that you are not interested in just being friends and that you don't want to go on walks, etc. Go no contact with her and concentrate on yourself. It will be painful, but you need to work through the pain. Otherwise, you will just be stuck in the same thoughts over and over again.
Good luck.
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