I don't think you have to stay overnight. It really would depend on what time of year you are cruising because, really, the only thing that usually interferes with a ships itinerary is a hurricane, otherwise, I think you would be fine to simply get off of one ship and then move right to the other. You may be different from me, but the thought of getting off of a ship and then finding transportation to a hotel, then settling in for a whole day, then repacking again and transferring again, etc. just sounds a bit exhausting.
Or, another alternative....if you truly think it would be boring to stay on the same ship, consider this, everyone who sails on the bigger ships state that there is no way to explore everything in one trip. Surely you can find a larger ship that sails two different itineraries to go B2B on.
My family has been B2B twice on the Carnival Elation, once out of Mobile and the next summer out of New Orleans, each time sailing the same itinerary and we never got bored. In fact, the time between cruises when we were the only five people left on the ship was the most exciting time of all.
Anyway, whatever you decide, I am sure that you will have an awesome time!