I don't know of any alternatives close to the port. The charge the last time we sailed from Tampa was $12 per day. (It's $15 per day now in Ft. Lauderdale, and Miami) You could probably park at the long term parking at the airport, and cab ride to the port, but I'm not sure if you'd come out enough ahead to make it worth while, or not.
If you're driving in and spending the night, some local motel/hotels have a plan where for a small fee, you can leave your vehicle on their property, and sail.
Give us a little more info, and maybe we can come up with something.
There is a Holiday Inn near the port that allows you to park there...and they run a free shuttle to the port. We stayed at the Hampton Inn...and they allowed you to park there for the week...but they did not have a shuttle. A cab only costs $17ish (before tip)...so that is a much cheaper option that parking at the port.
I'm heading out of Tampa for 7 days in a few weeks, and I feel that the price is manageable. Granted, would love for it to be free. However, I would feel much better leaving it in long term with security and a more closely watched eye. Never know though, things can still happen. But for me, it's worth it.
PS: Don't leave used vacation film in car. Almost always get's ruined.
I think the security at the long term parking garage in Tampa is pretty good now. They lock the garage up as soon as the ships sail. A few years back, before they started locking up, a whole bunch of high line SUV's were broken into in that garage, so they did indeed beef up security. It may be worth the $12 a day for the security, and being less than a block from walking aboard the ship.