Before the final payment date, you can definitely change the category of a reservation, as long as rooms in the new category are still available and the number of people in the room will fit in the lifeboat station for the new room. The fare you owe at final payment will be the fare for the new room as of the date of your initial booking. If you have already paid more than the cost of the new room, you will get a refund. If your initial room was Concierge level, you can move to a different category, but you will still owe at least the initial deposit for your Concierge room.
After final payment, you should still be able to move rooms, but I am not sure whether you could get a refund if the room cost goes down.