My wife and I are Newbies and we just got back from our first drift diving experience (in Coz). Everything everyone has said on this thread is correct. Our OW certification dives and following recreational dives were in the Floriday Keys. I believe your concerns, if like ours were, is as follows:
You probably did your check out dives with the boat anchored and you just pulled yourselves down that rope on descent and up it on ascents and safety stops. That is if you even did a safety stop in those dives. Many dive ops in the Keys doing check out dives do so in 25-30 ft. of water and the dives are usually no longer than 30 mins. I know, safety stops should be part of those dives, but the truth is they often aren't. But even if you did safety stops you probably held on to that anchor line at 15 ft. You won't have that line in Cozumel. So...the best advise I can give is the advise I got from so many divers when I got there. Make sure you have enough weight to get down and to hold you at the 15 ft. safety stop.
You will read opinions on BC control, proper weighting, etc. But I'm telling don't get into what my wife called "the competition." Here's where all the experienced divers reading this will rip me to shreads (especially if they have forgotten their newbie days or where just exceptional as newbies.)
If you need 22lbs to get down and hold a stop, use it. You will hear experienced divers whistle or shake their heads because they may think you should only be using 16lbs. However, they likely will never admit that they too were "overweighted" when they were newbies. Our Cozumel dive master admitted to me that nearly 100 percent of the new divers who go there dive overweighted. They are concerned about not have that anchor line to hand on to (again to descend or ascend). He totally understands this, but his concern (and appropriately so) was that new divers who are overweighted drag fins across the reefs. Just don't do it.
Even overweighted you can get enough buoyancy control (you do have a BC) to drift well ABOVE the reefs. You will gradually drop weights. I dropped 6lbs. in just the first 3 days. I dove six days total and by that 3rd day I was at the weight the DM suggested on the first day. The main thing new divers in Coz fear is shooting past that safety stop after an 80ft. dive. I got caught up in that "how much weight do you use" thing. I promise you...you will drop more weight as you become more confident. However, don't get too cocky. I tried to drop 4lbs on my second day there (3rd dive) -and yes, I couldn't hold that 15ft. stop for more than 30 seconds.
Here's a good tip from the dive master we had. If you're just not sure about holding that safety stop, stay with the DM if he is "staging" his ascent after inflating his marker for the boat. Our DM would reel out line and float the marker from about 60ft., hang there for a few minutes, and then continue to do the same thing as he slowly moved up the water column in stages until getting to 15'. In essence, you don't have to go directly from 70' to 15ft. In fact "staging" is becoming so popular that many divers believe it will soon be taught in OW, and if not, absolutely in AOW. I mean, our DM dives two or three dives just about every day of his life and he's never been bent. As in any discipline, classroom instruction usually comes from field applications. As any engineer will tell you, theories don't build roads and cities. It's learned first in the field and then taught in the classroom.
Don't fear drift diving. It's the easiest diving you will do. It's actually easier to maintain some decent level or buoyanyc control in a current. In an area with no current you might find you can't simply hold yourself in the water column without finning. We found that in a current, as in drift diving, the current is moving you and it was easier to look for that ideal buoyancy control. You can literally be completely vertical in the column and the current pulls you along without hardly moving a muscle.
Just relax and don't let this whole new diving experience overwhelm you. Just do everything the folks in this thread advise concerning choosing a dive op and safety and you'll have the best dives of your new diving life.