I've done that in the past, and there are so many factors that figure into whether you get a good deal or not,,,,and even whether you can get aboard or not. The good deals won't occur unless the ship isn't at capacity. Obviously, if it's sold out, you won't be able to get a cabin, at any cost. On the other hand, if she isn't sold out, and the line is scrambling to fill cabins, then, yes, you will get a "special deal" and it may well be very heavily discounted.
Years ago, we used to drive to the port, bags packed on the day of sailing, and get aboard, IF there were any open cabins for next to nothing,,,,however since 9/11 that can't be done anymore. (security you know)
If you're wide open to sailing,,,or not,,,then you might try the "last minute" thing. I'd rather pay a few bucks more, and know I'll be sailing.