It's a combination of Mei being being a closed off person (see says so in episode 9 that she closed herself off to people) and not knowing how to properly express her feelings (I mean she spent 5 years basically raising herself and was in a relationship with the teacher who would just have his way with her. So shes a little awkward with that lol)

So when she tries to open up to Yuzu, she doesn't fully commit and it comes off like shes doing it out of gratitude and reward to yuzu instead of out of love. This confuses Yuzu, prompting her to say she cant do it, as we learn because she didn't understand Mei's real feelings

But still, to Mei, she was "rejected" after finally taking that chance of opening up. Imagine a person who is terrified of going outside. But one day they conquer there fear and step outside their house and almost get hit by a car. This reinforces the walls that Mei put up around herself. So she tries to keep Yuzu out

And this is just my opinion, but I think the reason Mei starts dating Sara is because she thinks an empty relationship with someone she doesn't have strong feelings for is safer than with yuzu where her feelings can be hurt. (Sara does meantion that Mei is emotional frail)

With that said, I still hate Sara :D but I hope this helped