I never once got the impression that Saix felt any shame for what he was doing. His face doesn't emote at all in the scene.
Exactly, it's emotionless, almost like a robot. Saix' actions in Kingdom Hearts 2 (and the later half of Days) speak other things though as he acts like a total jerkass and is willingly conspiring with Xemnas.
Furthermore, there are several scenes where Xemnas is in the throne room speaking about internal Organisation matters and the only members present are Xigbar and Saix.
Axel noticed already in Days that something in Saix had deeply changed, and him having Xehanort-eyes already the first time you see him in Days gives quite a vibe that he was partly "norted" very early on.
Then there's this snippet:
Nomura said:
—By the way, Braig’s dealings with Master Xehanort in Birth by Sleep make sense, he was to become a vessel.
Nomura: There is a certain reason for Braig to proudly exclaim, “As for me, I’m already half Xehanort.” Isa (Saix) is included too. I think you’ll understand the details about their circumstances eventually.
It implies that Braig's and Isa's circumstances surrounding "the Nort incident" are similar if not the same.
It is true that Saix and Axel originally planned to betray Xemnas because they didn't trust them, but no one knows how long Saix really cared for that plan and at which point he only acted while already being in league with Xemnas.
It is presented to us in Days that Saix ordered Axel to eliminate Zexion because he was an obstacle to Saix' ascent, but Saix is already second-in-command and giving out missions
before the Castle Oblivion incident, so it might be possible that Saix' order to Axel was in fact just passed down from Xemnas to get rid of already known
"failed vessels" at Castle Oblivion, camouflaged by Saix to look as if he was still in league with Axel's and his plan.