Leia was raised as royalty, so of course she knows how to sound “proper”, but she’s not just some posh princess, so she also knows how to drop that act and be a normal person.
What about when she talks to han about the money near the end of the movie? Was that so she could come off as kinda snobbish or something
Yep. Notice it’s her line “If money is all that you love, then that’s what you’ll receive!” that gets the posh tone, but just before that, her line “It’s not over yet…” is very casual. It fits her character.