M.F.
Have you ever been on a cruise before? The day of embarcation is CRAZY and super rushed, even if you arrive the day before. Honestly, I would just hang at the hotel, swim, have a nice, leisurely breakfast and a quick lunch/late snack if time affords and get to the dock in plenty of time. It's going to be a long afternoon of waiting in lines and hauling luggage, so I wouldn't try to DO anything before you leave.
Ship diapers and toiletries to the hotel through Amazon. That will save a lot of time and trouble. Grab a bottle of Bourbon in the Duty-Free shop in the airport. The last thing I would want to be doing is trying to find a Target and a liquor store.
Frankly, since it's only going to be 24 hours on either end of the cruise, I'd ditch the rental car. See if the hotel offers an airport shuttle. See if the cruiseline offers or recommends shuttles to the docks. That way, you can also skip the need for carseats and save yourself a lot of money and hassles.