If I were you, I would stop asking them to give you some time to talk to you. I would now say to them that they have made their intentions known and you are now letting go of the relationship. If they see that you are willing to break it off, but they want to hold onto it, they will have to make the effort. But don't word it like an ultimatum, or hope that saying you're letting go will manipulate them into a positive response. Be clear and firm that it is over.
Because it sounds like it is over, so why put yourself in the position of begging for their attention, waiting for them to confirm it? You have enough to deal with. Take a stance.
Hot chick in the city.
Oh, oh, can't you see? Love is the drug for me.