I've been to Jamaica three times, now at least. I've stayed at Montego Bay, but it was at Jack Tar. Then a Beaches in Negril, and recently at the brand new Sandals Whitehouse resort on the south side--it's a very quiet and undeveloped area. I think we had a cruise stop there once before all that, so that's actually 4 times.
Yeah, it's a 3rd world country of a sort, but I've been mostly impressed with the locals--hard working, hospitable, and proud people, and a "real" culture. Yeah, I've had guys try to sell me pot in the marketplace, but what do you expect? I wouldn't wander downtown at night by myself, but I wouldn't do that in most American cities, either. Watch your back, no matter where you are.
Check out the south side of the island. Shopping trips are a bit of a hike, but it is quiet--although still plenty to do/eat/drink.
Every time my wife and I look into other places, we end up back at Jamaica based largely on price and because our experiences there have been very positive. I've been to Cayman, too, and it just seemed like a sterile tourist trap full of bankers. Jamaica is real.