When they start to think your embarrasing - what do you do?

DD just turned 10. she is particularly tired and emotional at the moment as they are having school swimming lessons. Increasingly she is saying "you embarrassed me today". For example today she was a bit late so I went to the office to get a late slip and took it to the classroom and it ended up she didn't need it so that embarrassed her that I came into the classroom when I didn't need to. so tomorrow she is starting a new activity and I can stay or watch or not, it doesn't matter. Normally I would stay and watch but I thought I might give her some space with a new group of people she doesn't know and just let her be there by herself.
So in general, do you keep behaving as you normally do and just ignore the fact that they might be embarrassed by you? Or do you let them have a bit more space and stay away from them so you don't get accused of embarrassing them.
Do all preteens/teens go through this stage?

Somewhere in between I think is good. It's good to give them space, but don't pull too far back.

I remember being alternately proud of and embarrassed by my mother as a teen, and I'm pretty sure my kids have the same capacity :) it's part of growing and developing a separate identity. It's also fairly individual to the child. My eldest daughter was much more independant than my eldest son.

For things like tomorrow when you can stay or go, ask her what she would like you to do. Tell her it's okay if she wants you to go, and that you are happy to stay - her choice. If it were me, I would pose the question as 'I have x,y,z to do - will I do it while you are at (activity) or do you want to come with me afterwards if I wait for you?' - something like that, that takes the pressure off, if she is feeling guilty about wanting space and makes it about what you are doing, not her, if that makes sense.

As for stuff like 'you came to the classroom when you didn't have to' - with my kids I'd probably laugh it off & tell them to get over it, to be honest. In a loving way. And you know your child - what is okay and not okay with her. But I don't buy in too much to the teenage (or in your case, pre-teen, which can be worse!) angst. Give them an inch and they will run a mile with that, lol.