Simcoediver has brought up a couple of great questions. My wife and I are on the Feb 9, 2019 Symphony sailing with the same itinerary. We cruise on Royal about 4 times a year. Lemme share my experience of cruising and diving.
Yes there are a lot of good reasons for bringing your own gear, I always do. I bring all the essentials and then I add gear to my rolling dive bag until it is about 10lbs shy of what the airline baggage maximum is. I leave a 10 lbs buffer in case the the gear is wet on the way back home or we happen to have bought more souvenirs than we expected. We always sail in a balcony room or above and drying my gear is an important factor. I also spend a lot of time out there on sea days.
My gear gets rinsed in the shower and dried on the balcony. The water onboard is quite salty (it is refined seawater) but it is close enough to freshwater for my gear. I bring along about 25' of cheap paracord and some clothespins. I make a temporary clothesline on Day 1 just below the wooden railing and above the glass panels. Finding attachment points has never been an issue on any of the 25 Royal ships we have ever been on. Hanging things here like bathing suits, booties, and rash guards is discrete and dries things out quickly. My BC, wetsuit, and regs get layed out on the table or on the back of a chair. I leave these things out there until they are dry and little more, then they are put back into the dive bag for the next port or the return flight home. If you leave items on your balcony for a long period of time you can get a subtle note from your room attendant "Notice: Please remove your personal items off the balcony, We are doing maintenance/cleaning in your area of the ship tomorrow". That maintenance/cleaning may or may not occur; but please take the hint: They don't want their ship looking like the outside of a dormitory.
As others have mentioned, you can dry your gear in your bathroom. There will be a retractable clothesline in there. The problem is there is no ventilation fan in any of the cabin bathrooms. So as another poster mentioned drying is dependent on the air conditioning. There likely won't be a lot of space in there either.
Part of divers gear bag should be small explosion proof box made by Pelican or Otterbox. If not just because it is waterproof and ideal for logbooks, cell phone, cash, passports, etc. It is also the perfect safe keeping spot for C cards, sunglasses, jewelry, or your ship ID card. A waterproof case is also a sand proof case--for those days at the beach. If you don't have a waterproof case as apart of your gear bag already--you need one.
Dive operators. Truth be told my wife and I spend a lot of time selecting a future cruise and it's itinerary. And on top of whatever ridiculous amount of time that is I spend another 3X that amount selecting what dive operator I want to spend my money with. Most times it turns into a spreadsheet. I factor in the expected ships time of arrival, who I might be able to get a 3rd or 4th dive in with, locations of the dive shops in comparison where the ship docks, previous experiences with different shops, what shop might offer a great or unusual location, what are the costs, reputations, and who is likely the ships excursion provider. Just finding all the dive operator opportunities other than the ones listed on PADI will take some time. I'll be honest, sometimes the ships excursion is the best option as much as I generally don't like to go that route.
So for our itinerary Simcoediver let me give you some helpful hints with dive operators at each port.
Nassau: You are diving with Stuart Cove and you have very, very, very limited second choices. Stuart Cove is the dive excursion provider for every cruise ship that sails into Nassau. And they are probably the largest dive provider on the planet and by most accounts do an excellent job. Can you save money by scheduling the dives on our own? Well, that will get a little questionable when you get on shuttle bus from the pier to the shop. Your two tank dive is going to cost $179 and be the most expensive you spend for any cruise ship diving.
Roatan: Anthony Keys is Royal Caribbean's dive excursion provider and has been in business for an impressive amount of years. There are a bunch of other options but.......?????
Costa Maya: I'm not sure who the current dive operator for Royal is. I was just there last December and I chose a 1 tank afternoon Lionfish Hunt with Octopus Divers. That's what my spreadsheet told me was my best option. Maybe the spreadsheet tells me something different for February 2019?
Cozumel: This is why I make spreadsheets. Seriously, are there not close to 100 dive operators in or near San Miguel??? I believe Sand Dollar Sports is the excursion operator for every cruiseship that docks in Cozumel. They are all about volume and diving with them has about a 98% chance of being a cattle boat. In addition, one of their dive boats operating as a Royal Caribbean/Celebrity dive excursion actually sunk during the excursion just a couple weeks ago. So; you have soooo many other good choices. I suggest you look no further than this: Your local dive shop probably has yearly trips to Coz. Who do they use? And, what better way to give back to your local shop then by asking their recommendation and then making your reservation specifically saying you were referred by the name of your local shop. That will give a pat on the back to your local shop and also tell whoever the dive operator is in Coz that your local shop is "doing the right thing".