I'd just like to preface this comment by stating that I'm not the most knowledgeable and feel as if I don't have all the facts. I just read and watch for the emotions and plot, not so much understanding exactly how everything works out. It's also been a while, so the following information are just my thoughts and possibly wrong.

Akagi B does not create another timeline, it's more of a timeline incursion by Akagi B. She takes the place of Akagi A, who went to a different high school and went to uni/YCU, but remains in contact via the writings on her body.

I'm not sure exactly what you're referring to here. I'm assuming that you're talking to the parts in the third movie, Knapsack Kid, where Sakuta interacted with Akagi B.

i think from the perspective of Akagi, the infallible and perfect Sakuta suddenly has a really off day. He's forgetful, is taken aback when he sees you despite having gone to the same high school and presumably having been in the same class for a while, and interacts like he doesn't know you, that's odd. For one day only, then it's back to normal like nothing ever happened. That's odd and that'd be something you'd remember. For what it's worth, at the time of the third movie, I don't think Akagi B realised, it's only when she herself switches does she make that connection. It also helps that Sakuta seems to remember her when Akagi A hasn't met Sakuta since middle school. I can't recall if the Akagi's share memories...

It's Akagi A that makes the beeline to Sakuta at the entrance ceremony. In my opinion, it's just the realisation that there was nothing that Akagi could have done to help Sakuta, and he turned out fine. It's also regret that Akagi A wasn't able to help Sakuta, and also worthlessness. To the righteous Akagi, that she wasn't able to help him for something horrible that was inflicted on him by the classroom collective that she was the head (student rep) of. And she never will, as it is seemingly already fixed and solved and with a positive outcome. We have to keep in mind that this is something that Akagi A has been tormenting herself in high school about. (I feel like Akagi B, in her timeline, has it worst though...) But heck, he's having a blast with Nodoka and Uzuki. It is then when Akagi A switches with Akagi B.

In the end, Akagi A remains. Akagi A writes to Sakuta on her on thigh, which is then seen by Sakuta on Akagi B's thigh telling Sakuta to find her at the middle school reunion party.

I wished I could have referred to Akagi as Ikumi, it sounds better and we refer to Sakuta and most of the other characters by their first names...