If Cortana was rampant, she would still be dying. Will, I remember reading somewhere that an A.I. choses what their hologram is going to look like and I specifically remember reading that Cortana did not want to have clothes because she didn't want to deal with the extra step in processing to give herself clothes. As far as the Domain goes, I wonder if the Domain is now imprinting itself as Cortana just because she is so recognizable to Chief and the UNSC. That being said, when Cortana was uploaded to the Domain, the information there was a two-way street. She gained information but she also transferred all of her information into the Domain. If the Domain now has control of Cortana, or what appears to be Cortana, that could definitely explain why she is not acting the same as the Cortana we all grew to love. When she uploaded herself to save herself, she gave the Domain access to all of the UNSC secrets. locations, ships, etc. This reminds me of why the Cole Protocol was written.
Who's to say Cortana's not dying? Like I said, I believe she only thinks she's not Rampant and safe now because that's what the Domain wants her to think. If she thinks she's not Rampant and not dying, then that allows the Domain to control her to a greater extent, because she won't be worry about her Rampancy or death. But, again, like I said, I think Cortana's actually dead, and that the Cortana we've interacted with in-game's an imprint (I'll use your word because it's better than whatever I said) on the Domain controlled by the Didact. I think the reason the Didact-Domain has specifically chosen Cortana as its "champion" is because the Didact-Domain recognized Cortana as being the "being" with the most intimate knowledge of the Librarian's true Reclaimer: John-117. I think this is echoed very well by Cortana reciting almost word-for-word lines the Didact spoke in Halo 4 and Escalation. As far as Cortana's appearance is concerned: Yes, AIs choose their own appearance, but it happens very early in the AI creation process, and AIs very, very rarely change their appearance afterward, because it can result in catastrophic matrix crashes. I don't know why Cortana chose to appear the way she did (I can't find anything on Halopedia about it, unless I'm just missing it, which is probably the case), but I believe she appears the way she does in-game is because the Didact-Domain wants her to appear as a Forerunner would've looked; the clothing she's wearing is very similar to what the Librarian was wearing in Halo 4. Otherwise, Cortana would have no reason to change her appearance.
[video]https://youtu.be/Mp9hzfTIJ7g[/video] I had no clue 343 brought a new lead writer for Halo 5. It makes sense, though. Halo 4's story was a lot better than 5. And while I don't agree with all of the points this guy makes in the video (his comparison to Black Team and Han and Chewie is completely off), I do agree that Brian Reed needs to go. It's for the betterment of Halo 6's story.