My family and I have been on a number of cruises over the past 5 years or so, as a matter of fact we were introduced to diving during a visit to Cozumel while on a Royal Carribbean cruise ship.
In regard to the tender situation, I have always booked dive trips during a port visit with the local providers. It is important to us to have our own fast boat and dive at our own speed and not be herded onto a boat of another 20 divers of varying degrees of proficiency. Swimming through silt for an hour because Jethro can't get his buoyancy right is not a vacation. That being said, I have also had to get off the ship quickly to meet the various dive ops.
Don't fall for the crap about not being able to get off on the first tender if you have not booked a ship excursion. They want the suckers that are overpaying for a crappy ship excursion to feel like they're special. (they are "special", but not in a good way) I have been 100% successful in getting off on the first tender after visiting the purser's desk on the ship with a copy of my confirmation info from my provider stating I needed to be at the dock at a certain time. It probably helps that I'm 6'5" and can't get a little loud....
My opinion is if they advertise port times on their itinerary, then it is their responsibility to make those ports available to me during those times. Obviously weather re-routes and such are beyond their control, but you get the idea. And if one purser gives you trouble, come back in an hour, there will be a fresh new one to deal with!
Good luck.
oh, and I concur about Christi at BlueXTsea, I'd start by contacting her in any case, if she can't help you she'll steer you to who can.