They do not issue a new KTTW card. They just write down the cnange for you. They do not "always" approve the change right there. Sometimes, they can issue you the change right then. Sometimes they take your cabin number and say that they will call you if the change can be done. Then, if they can do it, they call you before dinner seating starts.
Yes, if it is a change, it is for your whole party. If you get the change, you just need to remember your new table number or take the paper they give you when you go to dinner.