Yes, calling them directly will answer all those questions!
We use park and go in Fort Lauderdale for both cruises and flying out of FLL. Sign in online and you will advise when you need to parking for. From the ship port you just look for the shuttle bus. When you leave you tell them you're going to the airport.
They are pretty accommodating. Just a question as to how much it will cost.
I would call them and ask.
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