From my experience most places pay enhanced rates for Christmas. Quite often double time, or at least time and a half.
If your place of work pays enhanced rates for Christmas you'd hope they would include you in that, even though those are your set days.
To be fair, many homes operate a system where half the staff work Christmas Day, half Boxing Day so as to ensure everyone is equal and also has a chance at enhanced rates and that no one is working all the festive season days, which many consider unfair. Also many don't want to work at all over Christmas and want to be off (I never minded a morning or evening shift, but would not work a long day)
If you had wanted those days off, I'd have booked them off a lot further in advance as they have rotas and it can be hard to get cover at short notice. Depends what your boss says really, but it is very short notice for annual leave. They may not be able to grant it this late.
I suggest you are best off talking to your boss as they can tell you the homes policy (all vary). I'm a bit surprised you don't have it all set out already tbh as nearly everywhere I know of works out Christmas rotas etc well in advance and even if staff work set days, they still give them choices regarding it.
Good luck