Cayman's CITA guidelines which cover water-sports operators specifically prohibit solo diving, and any reputable operator will enforce this. Anyone who doesn't will also be making other shortcuts with the guidelines and safety. All of the shore diving sites I am aware of will most certainly enforce this, so shore diving is out unless you find a buddy or hire a guide. On weekends when the weather is nice the former is possible if you hang around some of the shore sites, but not guaranteed.
As a solo traveler, you best bet is boat diving, and go with the guide. Many people on SB trip over themselves to recommend their favorite boat, but I'd advice you to look at the operator's website to see what sort of boats they are using first. Some of the boats here in GC offer small groups, but the boats themselves are small, offering little in the way of space, shade, or amenities. Check the operator's website to see photos of the boat first.
I'll second Saboteu's recommendation for Comfort Suites or Holiday Inn. That's about as economical as it gets here, unless you find a good AirBNB.