Hi...there's more rental car companies than just Avis that you can rent from at the airport and drop off at the port, so you shouldn't have a problem reserving a car with one of them.
We use Budget...my husband gets a corporate discount through them, but if you belong to BJ's or Costco you can get discount coupons through them for several rental car companies.
I booked our Budget rental car in February of this year for our September trip with no problem.
Fortunately for you, I don't think the time you are traveling is that busy a time. For us, travelling between Christmas and New years... most of the cars are already booked...
The Budget office near the port is at 8401 Astronaut Blvd. They provide a complimentary shuttle from their office to the ship. It's only a 5 min. ride.
Budget is <i><b>almost</b></i> as convenient as Avis... but I don't usually mention them because their rates tend to be significantly higher than Avis, Hertz, or National.
If you have a corporate discount like Carol RN does, you may want to consider them as well.
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