We did a cruise in March from Fort Lauderdale (Port Everglades). Yes, there is ample parking! We were much later in arriving than we wanted to be (around 1:30pm) and there still was plenty of parking. They give you a little tag to hang on your car, and when we left we paid...can't find the receipt but I believe it was $120.00 for the week. Super convenient though.
As for better parking lots, they ask us what ship we were on and directed us to a specific lot (there are many lot numbers) based on what ship we were sailing (we were going on the Westerdam). They have certain areas set aside for passengers of certain ships, basically right alongside the ship.
The only stressful part was getting past the Fort Lauderdale "strip" itself. Very crazy busy and a little frustrating lol.