Yes, the Sunset handles checked baggage between LA and New Orleans. At most stations, Amtrak will accept checked baggage up to 24 hours in advance. LA will not having a problem accepting checked baggage for the Sunset around noon or even earlier the same day.
Note you will not have access to any checked bags during the the trip. If you want to lighten yourself of any carry ons and you are in a sleeper, you can leave your carry ons with the staff at LA's Metropolitan Lounge. They will either keep it behind the desk or put it in conference room they have. If you aren't in a sleeper you can day check your carry ons with the baggage room for $10/bag.
Note you will not have access to any checked bags during the the trip. If you want to lighten yourself of any carry ons and you are in a sleeper, you can leave your carry ons with the staff at LA's Metropolitan Lounge. They will either keep it behind the desk or put it in conference room they have. If you aren't in a sleeper you can day check your carry ons with the baggage room for $10/bag.