I'm sure you'll like Morritt's so long as you can kick back and relax. Morritt's is on the
very quiet East End of Grand Cayman.
There are only three dive ops actually in East End as opposed to 30+ or so scattered around Georgetown, Seven Mile Beach, and West Bay on the west side of the island. East End dive sites are a bit healthier as a result.
In East End you can choose from Cayman Dive Lodge, Ocean Frontiers, or Tortuga Divers. As others have pointed out, I work at Tortuga Divers which is located at Morritt's Tortuga Club where you'll be staying.
AFAIK, all the East End ops can teach an Advanced Course. I believe that all would teach a PADI course. Such a course would require a deep dive, a navigation dive, and three additional dives chosen from a selection of topics.
Shore diving is very popular on Grand Cayman. In fact I went on a shore dive up at Divetech today (Yes, that's what this dive instructor does on his day off.) and saw a great variety of Tarpon. It is a lovely dive, but about as far as you can possibly be from Morritt's. I got stuck in bad rush hour traffic and it took 1 hour and 45 minutes to make the 20 mile drive home.
There are a few spots for shore diving East End, all along the north shore. By October we may be seeing ENE winds which might make entries and exits dicey. Ask in any of the EE dive shops about suitable sites for a given day.
There are many more shore diving sites on the west side that will be sheltered. Sunset House, Seaview, Eden Rock and Devil's Grotto are popular sites just south of Georgetown. Turtle Reef and Cobalt Coast in West Bay (both are Divetech shops) are very popular too. All of these sites have a dive shop on site where you can rent tanks and weights.