The blood princess has decorum and conditioning that the knight is not bound by. Further both personalities realize that sin lord worship is bad for everyone involved.
more specifically I'd say bai ji feels like they can forget everything that happened when they're in human form, but in vampire form, they can't forget at all.
at least that's my interpretation- the whole training thing was so traumatic bai ji tries their best to pretend it never happened, but obviously that's impossible when they're in vampire form, because that's a constant reminder, it's gotten better though, note that they're having easier and easier times speaking and acting normally in vampire form to some extent
the knight and the vampire are basically separate people
no they aren't. that's a misconception that the manga implies, but even in the manga, there's no change. it's just trauma