It is possible (though of course, not guaranteed) that cabins will be available on the morning of embarkation. After you check in, you can go to the upgrade desk to check. If no rooms are available, there will be a sign saying something like "Captain Mickey reports that his ship is sailing full. No upgrades are available." If upgrades are available, you can ask what your cost would be to have one. The cost will be based on the level of the room you originally booked. This cost is often, but not always, significantly lower than the cost of upgrading a week before sailing. If there are upgrades to rooms at levels above the level you want, it wouldn't hurt to check back before boarding to see if anyone has upgraded and freed up cabins.
Unless you have purchased room decorations, changing your room will create minimal trouble for room stewards, since they will need to clean all rooms completely anyhow. Even if you do have a special set-up in your room, I would guess that it can be moved to the new room, though I have never tried that. My family did a port upgrade in Copenhagen, and our dining tickets and luggage were delivered to our new room with no problems.
Good luck!
