Of course there are drawbacks to being the closest dive to the cruise ship port. Consider taking a cab north to Sundivers at Turtle Reef where fewer people will go. Dive op on site with very nice shore diving, well known restaurant upstairs (Cracked Conch), bar to hang out after, and near the Turtle Farm if you want to check that out. You can make a full day of the stuff there.
You could also go to Sunset House, a bit further south of Eden Rock just getting out of town. You wouldn't want to walk with dive gear, and I don't think the diving there is as nice anymore as Turtle Reef. But decent enough with some variety, including their well known mermaid statue, or an UW rendezvous with one of the subs if you time it right and they're still running by there. Also great bar/restaurant on the water to hang out at.
Best of all would be if your schedule allows meeting up with one of the smaller ops and doing a trip to the North wall. The walls are what Cayman is known for and while there's some places you can swim to the wall from shore, it's a long haul. On a cruise ship excursion you may go to a site on the West wall, but they're well dove and not what they used to be.
Stingray City is of course famous. One of those things worth doing once, but I'm not sure if I would give up a chance at some good wall dives for it.